Modern Physics

Step 1: De Broglie relation.
where = mass, = charge, = accelerating potential.
Step 2: For proton.

Step 3: For -particle.
Mass , charge .
Step 4: Ratio.
If approximated as integer ratio: .
Step 5: Conclusion.
The ratio .

Step 1: Recall de Broglie relation.
According to de Broglie, a moving particle has wavelength:
Step 2: Substitute momentum.
So,
Step 3: Conclusion.
The de Broglie wavelength is inversely proportional to momentum.

Step 1: In solids, atomic energy levels overlap due to the close proximity of atoms, forming continuous ranges called energy bands. Step 2: The two main types of bands in a solid are the valence band (occupied by electrons) and the conduction band (where free electrons exist). Step 3: The gap between these bands determines the electrical properties of the material.

Step 1: The photoelectric effect is the phenomenon in which light incident on a metal surface ejects electrons. Step 2: Albert Einstein explained this effect by proposing the quantization of light energy in the form of photons. Step 3: This discovery confirmed the particle nature of light and won Einstein the Nobel Prize in Physics in 1921.

The instantaneous power dissipated in an AC circuit is given by the formula:
Where:
- is the maximum voltage,
- is the maximum current,
- is the phase difference between the voltage and current.
From the given expressions:
- , so volts,
- , so mA or A,
- The phase difference .
Now, substitute these values into the power formula:
Since , we get:
Thus, the correct answer is option (B) 2.5 watt.

Young's double slit experiment provides strong evidence for the wave nature of light. It demonstrates the phenomenon of interference, which occurs when light waves from two slits overlap and produce constructive or destructive interference. The experiment shows that light has a wave-like nature, as interference patterns are observed.
Now, for two different wavelengths, the condition for constructive interference (bright fringes) for each wavelength is given by the equation:
Where:
- is the fringe separation (distance between adjacent bright fringes),
- is the fringe number,
- is the wavelength of the light,
- is the distance between the screen and the slits,
- is the distance between the slits.
For two wavelengths, the bright fringes will coincide when the fringe separation for both wavelengths is equal, i.e., when:
Thus, we need to find the minimum distance where the first bright fringe for both wavelengths coincide. Let’s calculate it.
We are given:
- ,
- ,
- ,
- .
To find the minimum distance, we equate the fringe separations for both wavelengths:
Simplifying, we find:
Substituting values:
Solving for :
Thus, the minimum value of and that satisfies the condition is when and . Therefore, the minimum distance from the central maximum where the bright fringes for both wavelengths coincide is given by:
Thus, the minimum distance is 3 cm.

Capacitance of a Parallel Plate Capacitor: A parallel plate capacitor consists of two conducting plates separated by a distance , with area of each plate. The capacitance of such a capacitor is given by the formula: where: - is the capacitance,
- is the permittivity of free space,
- is the area of one of the plates,
- is the separation between the plates.
Effect of Dielectric Medium: When a dielectric medium with dielectric constant (also known as relative permittivity) partially fills the space between the plates, the capacitance is modified. The dielectric constant increases the capacitance by a factor of . If the dielectric only fills part of the space between the plates, the system can be treated as two capacitors in parallel: 1. One part of the capacitor (with area ) is filled with the dielectric. 2. The other part (with area ) is filled with air or some other dielectric with permittivity .
Capacitance of the Region with Dielectric: The capacitance for the part of the capacitor with dielectric is: where: - is the area of the plates covered by the dielectric,
- is the distance between the plates covered by the dielectric.
Capacitance of the Region with Air: The capacitance for the part of the capacitor with air (or vacuum) is: where:
- is the area of the plates not covered by the dielectric,
- is the distance between the plates not covered by the dielectric.
Total Capacitance: Since the areas and add up to the total area , and the distances and add up to the total distance , the total capacitance is the sum of the capacitances of the two regions: Formula for Capacitance: For the case where the dielectric partially fills the space between the plates (i.e., ), the final formula for the capacitance becomes: This is the general formula for the capacitance of a parallel plate capacitor partially filled with a dielectric medium.

Huygens' Principle states that every point on a wavefront can be considered as a source of secondary wavelets, and the position of the wavefront at any later time is the surface tangent to these secondary wavelets. In other words, each point on the wavefront acts as a secondary source of waves, and the new wavefront is formed by the envelope of all these secondary waves.
This principle can be used to explain the laws of reflection of light:
1. Law of Reflection (First Law):
The angle of incidence ( ) is equal to the angle of reflection ( ), i.e., This is confirmed by Huygens' principle because, when the wavefront strikes a reflective surface, each point on the wavefront produces secondary wavelets. These wavelets propagate and form a new wavefront that follows the law of reflection.
2. Law of Reflection (Second Law):
The incident ray, the reflected ray, and the normal to the surface of reflection all lie in the same plane. This can be understood by imagining the wavelets from the incident wavefront interacting with the surface, forming a new wavefront that obeys this geometrical constraint.
Thus, Huygens' principle provides a clear explanation for the reflection of light, where the wavefront interacts with the reflecting surface, and the reflected wavefront is formed by the secondary wavelets.

Unpolarized light is light that oscillates in all directions perpendicular to the direction of propagation. Polarized light, on the other hand, only oscillates in one direction. The process of polarizing light involves passing it through a polarizing filter, which only allows light oscillating in a specific direction to pass through.
The diagram below shows the difference between polarized and unpolarized light:
In the case of the polarized light, the oscillations are restricted to one plane, while in unpolarized light, the oscillations are in multiple planes.
Now, the problem gives the polarizing angle for a transparent medium as . The polarizing angle is related to the refractive index of the medium using Brewster's Law, which is given by: where is the polarizing angle. Substituting the given value of : Since , we have: Thus, the refractive index of the medium is .
Next, we can find the angle of refraction using Snell's Law, which is: where and are the refractive indices of the two media, is the angle of incidence, and is the angle of refraction. Since the angle of incidence is equal to the polarizing angle , and we know the refractive index of air is approximately 1, we can use: Solving for : Thus, . Therefore, the angle of refraction is .

An astronomical telescope is a type of optical telescope used to observe distant celestial objects. It consists of two lenses: the objective lens and the eyepiece. 1. Objective Lens:
The objective lens gathers light from a distant object and forms an intermediate image at its focal plane. 2. Eyepiece Lens:
The eyepiece lens magnifies the intermediate image, making it visible to the eye. The final image is formed at infinity, meaning it is highly magnified and can be observed clearly without straining the eyes. 3. Ray Diagram:
In the case where the final image is formed at infinity, the object is placed at a distance greater than twice the focal length of the objective lens. The rays from the object are parallel and converge to form an image at the focal plane of the objective lens.
The eyepiece lens then produces an image that appears at infinity. The ray diagram for an astronomical telescope when the final image is at infinity is as follows:
- The objective lens produces a real, inverted image at the focal plane.
- The eyepiece lens magnifies this real image and makes it appear as an inverted image at infinity.

An equipotential surface is a surface on which the electric potential is constant at every point. It is defined such that no work is done in moving a charge along the surface because the electric potential difference between any two points on the surface is zero. The electric field is always perpendicular to the equipotential surface.
To answer the question, we need to consider the implications if two equipotential surfaces were to intersect.
1. Electric Potential at a Point:
- The electric potential at a point in space is a scalar quantity, meaning that at any given point, the potential has a single value.
- If two equipotential surfaces were to intersect, it would imply that a point would have two different potential values (since each surface corresponds to a different potential). This is a contradiction because a point cannot have two distinct potential values at the same time.
2. Electric Field Direction:
- The electric field is always perpendicular to the equipotential surface. If two equipotential surfaces were to intersect, the direction of the electric field would have to be ambiguous at the point of intersection because there would be two perpendicular directions corresponding to the two surfaces. This would violate the definition of the electric field.
3. Conclusion:
- Therefore, two equipotential surfaces can never intersect. If they did, it would imply a violation of the uniqueness of electric potential at any given point in space, and it would lead to an undefined situation for the electric field.
Thus, two equipotential surfaces cannot intersect.

The electrical capacity, or capacitance, of a conductor refers to its ability to store electric charge when a potential difference is applied across it. The capacitance of a conductor is defined as the amount of charge it can store per unit potential difference . Mathematically, it is given by the equation:
Where:
- is the capacitance in farads (F),
- is the charge stored on the conductor in coulombs (C),
- is the potential difference across the conductor in volts (V).
The unit of capacitance is the farad (F), which is defined as the amount of charge that can be stored per unit potential difference. In terms of SI units, we can express the farad as:
This means that if a conductor has a capacitance of 1 farad, it will store 1 coulomb of charge when a potential difference of 1 volt is applied across it.
The electrical capacity (or capacitance) depends on the physical characteristics of the conductor, such as its surface area, shape, and the dielectric properties of the material surrounding it. For a parallel plate capacitor, for example, the capacitance is given by:
Where:
- is the capacitance,
- is the permittivity of free space ( ),
- is the area of the plates,
- is the distance between the plates.
Thus, the electrical capacity of a conductor is the ratio of the charge stored to the potential difference, and its unit is the farad (F).

Resistivity ( ) is a property of the material itself and does not depend on the shape or size of the wire. It is determined by the material through which the current flows. The formula for resistivity is: where: - is the resistance,
- is the resistivity,
- is the length of the wire,
D - is the cross-sectional area of the wire.
Thus, resistivity depends on the material, and it does not change with the length or thickness of the wire.

The phenomenon of the photoelectric effect supports the particle nature of light. When light of sufficient frequency (threshold frequency) is incident on a metal surface, electrons are emitted from the surface. This phenomenon could not be explained by the wave theory of light, but Einstein's quantum theory of light provided a clear explanation. According to this theory, light consists of discrete packets of energy called photons. When a photon strikes the metal surface, it imparts its energy to the electron, causing the electron to be ejected from the surface.
The Einstein equation for photoelectric emission is given by: Where: - is the energy of the incoming photon,
- is Planck's constant,
- is the frequency of the incident light,
- is the work function of the metal,
- is the kinetic energy of the emitted electron.
Laws of Photoelectric Emission:
1. Emission of Electrons: When light of frequency greater than or equal to the threshold frequency strikes a metal surface, electrons are emitted.
2. Effect of Intensity: The intensity of light affects the number of electrons emitted, not their energy. Higher intensity leads to more emitted electrons.
3. Effect of Frequency: The energy of emitted electrons depends on the frequency of the incident light. Electrons are emitted only if the frequency of light is above the threshold frequency, regardless of intensity.

The electric potential energy of a system of point charges is the work required to assemble the charges from infinity to their given configuration. It is mathematically expressed as where and are the charges and is the distance between them. In the given figure, four charges are placed at the corners of a square of side , with at the top left, at the top right, at the bottom right, and at the bottom left. For the interactions between adjacent charges separated by a distance , each pair consists of unlike charges, so the potential energy contribution from each side is . Since there are four sides, the total contribution from adjacent charges is . Next, consider the interactions along the diagonals, where the distance between charges is . Along one diagonal, we have two like charges, giving a positive contribution of each, while along the other diagonal, the charges are unlike, giving a negative contribution of each. When summed, these diagonal contributions cancel out to zero. Therefore, the total electric potential energy of the system is obtained by adding the adjacent and diagonal contributions: Hence, the total potential energy of the system of charges arranged at the corners of the square is negative, equal to .

To achieve the condition of minimum deviation in a prism, the following conditions must be met:
1. The incident ray and the refracted ray inside the prism must make equal angles with the prism surface. This means that the angle of incidence at the surface must be equal to the angle of refraction inside the prism.
2. The angle of deviation is minimized when the ray passes symmetrically through the prism. This implies that the path of the light through the prism should be symmetric in nature with respect to the prism's surface.
3. The angle of incidence at the silvered surface (the reflecting surface) must allow total internal reflection. The light should reflect off the silvered surface in such a way that the ray retraces its original path inside the prism. This occurs when the angle of incidence on the surface is greater than the critical angle for total internal reflection.
The relationship between the angles is given by: where: - is the angle of incidence on surface , - is the angle of refraction inside the prism, - is the refracting angle of the prism. Also, the angle of deviation is given by: For minimum deviation, the deviation is at its smallest value when the ray enters and exits the prism symmetrically. This means that the angle of incidence and the angle of refraction must be equal, i.e., . Thus, for minimum deviation: Now, considering the silvered surface at the other end of the prism, the ray must undergo total internal reflection. The critical angle for total internal reflection is given by: where is the refractive index of the prism. Therefore, the critical angle is: For total internal reflection to occur, the angle of incidence at the silvered surface must be greater than this critical angle. Thus, we set the angle of incidence such that the reflected ray retraces its path. The angle must satisfy: Thus, the angle of incidence on the surface is 45° for the reflected ray to retrace its path after total internal reflection at the silvered surface.

The given circuit consists of resistors of values , , and , with a voltage difference between points and . We need to find the equivalent resistance when: - (i) , - (ii) .
For this circuit, when , the resistors will be connected in series, and the equivalent resistance is the sum of the individual resistances. Thus, the equivalent resistance will be:
For the second case, when , the resistors will be in parallel. The equivalent resistance for parallel resistors is given by the reciprocal sum of the individual resistances:
Solving this:
Thus, the equivalent resistance is:

Capacitor Combination: The given combination of capacitors is shown below:
1. Step 1: Calculate the Equivalent Capacitance of the 4 μF and 5 μF Capacitors in Series:}
For capacitors in series, the equivalent capacitance is given by: Substituting the values: Thus: 2. Step 2: Find the Equivalent Capacitance of the Entire Combination:}
Now, the equivalent capacitance of the 3 μF capacitor in parallel with the series combination (2.22 μF) is: So, the total equivalent capacitance of the combination is 5.22 μF.
3. Step 3: Calculate the Total Energy Stored:}
The energy stored in a capacitor is given by the formula: Substituting the values ( and ): Thus, the total energy stored in the combination is 10.44 μJ.

In the given circuit, we have four capacitors , , , and connected in a combination of series and parallel. The total voltage across the combination is given as . Let’s break down the solution into steps:
Step 1: Simplifying the Combination of Capacitors
First, we need to combine capacitors , , and in parallel. The total capacitance for parallel combination is given by: Substituting values: Step 2: Combining with in Series
Now, we combine the equivalent capacitance with in series. The equivalent capacitance for capacitors in series is given by: Substitute the values: Thus, the equivalent capacitance of the combination is approximately .
Step 3: Charge on Capacitor
The charge on the capacitor is given by: Substitute the values: So, the charge on capacitor is .
Step 4: Total Stored Energy
The total stored energy in the combination of capacitors is given by: Substitute the values: Thus, the total stored energy is .

Let the capacitance of each capacitor be and the total voltage of the battery is . Since the capacitors are connected in series, the equivalent capacitance of the series combination is given by: Thus, the equivalent capacitance is: Now, using the relation for voltage and charge in a capacitor, the potential difference across each capacitor is: Therefore, the potential at point is 20 V (since it is connected directly to the battery) and the potential at point is 0 V (since it is earthed). Thus, the potential at is 20 V and at is 0 V.

Rutherford's -Particle Scattering Experiment:
In Rutherford's experiment, a beam of -particles was directed at a thin gold foil. By observing the scattering angles of the -particles, Rutherford concluded that most of the mass of an atom is concentrated in a tiny, dense nucleus at the centre, and that the majority of the atom's volume is empty space. The -particles were deflected at large angles by the positive charge concentrated in the nucleus. This led to the discovery of the nuclear model of the atom, where electrons orbit around a dense nucleus. Shortcomings of Rutherford's Model:
Rutherford's model, while groundbreaking, had certain flaws:
- It did not explain the stability of the atom. According to classical electromagnetism, the revolving electrons would emit radiation, causing them to lose energy and spiral into the nucleus, leading to atomic collapse.
- It could not explain the observed spectral lines of hydrogen, as predicted by experimental data, where the electron should continuously radiate energy and not remain in discrete energy levels.
Rectification by Bohr's Model:
Niels Bohr improved upon Rutherford's model by introducing the idea that electrons exist in discrete orbits or energy levels. The electrons can only occupy these stable orbits without radiating energy. Radiation occurs only when an electron jumps from one orbit to another. The energy associated with these orbits is quantized, and Bohr used this idea to explain the discrete spectral lines observed in hydrogen.

The surface charge density is the charge per unit area on the surface of a conductor. It is given by the formula:
Where:
- is the charge distributed over the surface,
- is the surface area of the sphere.
For a sphere, the surface area is given by:
Thus, the surface charge density for a sphere of radius is:
The unit of surface charge density is:
Now, we are given two concentric hollow spheres with radii and , and their surface charge densities are equal. This implies:
Thus, we have the following equations for the surface charge densities of both spheres:
Since , we can equate these expressions:
This simplifies to:
However, this is not possible if , implying that the surface charge densities cannot be the same unless . Electric Potential at the Common Centre Now, let’s calculate the electric potential at the common centre (which is the centre of both spheres). The electric potential at the centre of a spherical shell due to a charge distributed uniformly on its surface is given by:
For a concentric configuration of two spheres, the potential at the common centre will be the sum of the potentials due to both spheres. Therefore, the total potential at the common centre is:
Thus, the electric potential at the common centre of the spheres is:
Where: - is the charge distributed on the spheres,
- and are the radii of the spheres,
- is the permittivity of free space.

The electric flux linked with a surface is given by the dot product of the electric field and the area vector :
The area vector is given as , and the electric field is . The dot product of the electric field and the area vector is:
Since the dot product of two perpendicular vectors is zero and the dot product of two parallel vectors is the product of their magnitudes, we get:
Thus, the electric flux linked with the surface is .

The refraction of light through a single spherical surface is governed by the formula for refraction at a spherical surface, given by:
Where:
- and are the refractive indices of the media (in this case, for air and for glass),
- is the object distance (the distance of the point source from the spherical surface),
- is the image distance (the distance of the image formed),
- is the radius of curvature of the spherical surface.
Given: - Radius of curvature ,
- The object is placed in air, so ,
- The refractive index of glass .
Substituting these values into the formula:
Thus, the image will be formed at a distance from the spherical surface. To find the position and nature of the image, the value of (object distance) must be specified, as it determines the final image location.
The ray diagram would show light rays entering the spherical surface from the object at a distance , refracting and converging at the image point .

Diffraction through a Single Slit:
In the case of diffraction of light through a single slit, the central maximum is formed on the screen due to constructive and destructive interference. The angular width of the central maximum is given by the angle for the first minimum. The condition for the first minimum is:
Where:
- is the width of the slit,
- is the wavelength of the light,
- is the order of the minima.
For the central maximum, we consider the case of , and the angular width of the central maximum is the difference in angles between the first minima on either side of the central maximum:
The angular width of the central maximum is twice the angle for the first minimum:
For small angles (which is usually the case in most practical diffraction experiments), , so the formula for the angular width becomes approximately:
This formula gives the angular width of the central maximum. The linear width on the screen at a distance from the slit is given by:
Thus, the width of the central maxima on the screen is .
The intensity distribution of light through a single slit diffraction experiment shows a central bright fringe (central maximum) with diminishing intensity for subsequent maxima and minima.

When an electric field is applied across a metallic wire, the free electrons experience a force that causes them to move in the direction opposite to the electric field, resulting in a drift velocity . The relationship between the drift velocity and the electric field can be derived using the following steps:
Step 1: Force on an electron due to electric field.
The force on an electron of charge due to the electric field is given by:
Where:
- is the force on the electron,
- is the charge of the electron ( ),
- is the electric field.
Step 2: Acceleration of an electron.
From Newton’s second law, the acceleration of the electron is given by:
Where:
- is the mass of the electron.
Step 3: Drift velocity.
The drift velocity is the average velocity that the electron acquires due to the electric field. This velocity is reached after a characteristic time called the relaxation time , which is the average time between collisions of the electron with the atoms in the wire. The drift velocity is related to the acceleration by:
Thus, the drift velocity of the electrons is given by:
This equation gives the relationship between the drift velocity , the electric field , and the properties of the electron (charge and mass).

Total internal reflection occurs when light hits the boundary between two media at an angle greater than the critical angle and gets entirely reflected back into the denser medium. This phenomenon plays a crucial role in several important optical applications. Two common applications are: 1. Optical Fibers:
Optical fibers use total internal reflection to transmit light signals over long distances. The core of the fiber has a higher refractive index than the surrounding cladding, allowing the light to be continually reflected inside the fiber core. This process minimizes signal loss, making optical fibers ideal for telecommunications, internet data transfer, and medical endoscopy.
2. Prism in Periscopes and Binoculars:
Total internal reflection is used in the design of optical instruments such as periscopes and binoculars. In periscopes, prisms are used to reflect light through 90-degree angles without any loss of light intensity. This allows a viewer to see over obstacles by redirecting light from the scene into their eyes. Similarly, in binoculars, prisms help in the efficient reflection of light to give a clear and bright image.

A galvanometer can be converted into a voltmeter by connecting a high resistance in series with it. This series resistance is chosen such that it allows only a small current to flow through the galvanometer when a high potential difference is applied. The total resistance of the voltmeter is then the sum of the resistance of the galvanometer and the series resistance:
This modification ensures that the voltmeter can measure high voltages without damaging the galvanometer or causing excessive current flow.
The full-scale deflection of the galvanometer will correspond to the full-scale voltage of the voltmeter, which can be calibrated accordingly.

An astronomical telescope is designed to view distant objects in the sky, such as stars and planets, at a magnified scale. It consists of two lenses: the objective lens and the eyepiece.
- The objective lens ( ) has a large focal length and is placed at a distance from the object. It forms a real, inverted image of the object at its focal plane.
- The eyepiece ( ) is used to magnify the image formed by the objective lens. It is placed close to the focal plane of the objective lens and is used as a magnifying glass.
In the diagram: - is the objective lens.
- is the eyepiece.
- and are the focal points of the objective and eyepiece, respectively.
- The image formed by the objective lens is at the focal plane , and it is magnified by the eyepiece.
The magnifying power of the telescope is the ratio of the angular size of the image seen through the telescope ( ) to the angular size of the object when viewed with the unaided eye ( ). The formula for magnifying power of the telescope is:
Where: - is the focal length of the objective lens,
- is the focal length of the eyepiece.
The magnifying power is directly proportional to the focal length of the objective lens and inversely proportional to the focal length of the eyepiece.

A bi-convex lens is a type of lens in which both surfaces are convex. The focal length of a lens is determined by the lensmaker's formula:
Where: - is the focal length of the lens,
- is the refractive index of the material of the lens,
- and are the radii of curvature of the two surfaces of the lens.
For a bi-convex lens, (convex surface) and (concave surface). The refractive index of the glass is .
Substituting these values into the lensmaker’s formula:
Thus, the focal length of the lens is:
The lens is a converging lens because it is bi-convex.
Now, when the lens is filled with a liquid of refractive index , we use the modified lensmaker's formula:
Where is the refractive index of the liquid. Substituting the values into the formula:
Thus, the new focal length of the lens when filled with liquid is:
The nature of the lens remains converging, but its focal length decreases due to the increased refractive index.
The power of the lens is given by:
The power in air is:
The power in the liquid is:

In optics, the polarising angle is the angle of incidence at which light reflected from a surface is completely polarised. The critical angle is the angle of incidence at which total internal reflection occurs when light passes from a denser to a rarer medium.
Brewster’s law gives the relation between the polarising angle and the critical angle , which is:
This relationship arises because at the polarising angle, the reflected and refracted rays are perpendicular to each other, and the angle of refraction is the critical angle. The derivation uses the fact that the refracted light at the critical angle makes an angle of with the reflected light.
Thus, the correct relationship between the polarising angle and the critical angle is:
Hence, the correct answer is option (A) .

The mobility of a charge carrier is defined as the ratio of its drift velocity to the applied electric field . Mathematically, this is expressed as:
Where: - is the drift velocity of the charge carrier, - is the electric field applied.
Mobility is a measure of how quickly a charge carrier can move through a medium under the influence of an electric field. Therefore, the correct relationship is , which makes option (A) the correct answer.
Hence, the correct answer is option (A) .

When two cells with different electromotive forces (emfs) and internal resistances are connected in parallel, the equivalent emf and the equivalent internal resistance can be found using the following approach:
Step 1: Calculate the equivalent internal resistance.
The two internal resistances and are in parallel, so the total internal resistance is given by:
Therefore, the equivalent internal resistance is:
Step 2: Calculate the equivalent emf.
The equivalent emf of two cells in parallel is given by the formula:
This formula takes into account both the emfs and the internal resistances of the two cells. The equivalent emf is a weighted average of the two emfs, where the internal resistances act as the weights.
Thus, the formula for the equivalent emf of the combination of cells is:

Intensity of Electric Field on the Bisector of an Electric Dipole:
Consider an electric dipole consisting of two charges and , separated by a distance . The dipole is placed along the -axis, with the charges located at and . The point where we need to calculate the electric field lies on the bisector (the equatorial line) of the dipole, which is at a distance from the center of the dipole. The angle between the line joining the charges and the point on the bisector is . To calculate the electric field at this point: 1. Electric field due to the positive charge : The electric field due to a point charge is given by: For the positive charge at a distance from the point, the electric field is directed radially away from the charge. Since the point lies on the bisector, the field due to the positive charge will have a component along the direction of the bisector. 2. Electric field due to the negative charge : Similarly, the electric field due to the negative charge at the same distance will also have a component along the bisector, but it will be directed towards the negative charge. 3. Net Electric Field on the Bisector: Since both electric fields are of equal magnitude but opposite directions, they add up along the bisector. The net electric field is: where is the dipole moment of the system. Thus, the intensity of the electric field on the bisector (equatorial line) of an electric dipole is:

Capacitance of a Parallel Plate Capacitor with Partial Dielectric Slab:
Consider a parallel plate capacitor with plate area and separation between the plates. Let the capacitor be partially filled with a dielectric material of dielectric constant and thickness . The remaining gap is filled with air. The total capacitance can be considered as the combination of two capacitors in series: 1. The capacitor with dielectric slab: Capacitance , 2. The capacitor with air gap: Capacitance . 1. Capacitance with dielectric slab: The capacitance of the part filled with the dielectric is given by: 2. Capacitance with air gap: The capacitance of the part filled with air is given by: 3. Total Capacitance (Series Combination): Since the two capacitors are in series, the total capacitance is given by: Substituting the expressions for and : Simplifying the expression: Thus, the total capacitance is: So, the capacitance of a parallel plate capacitor with a partial dielectric filling is the same as that of an empty capacitor, but with the dielectric slab affecting the effective area.

Principle of a Moving Coil Galvanometer:
A moving coil galvanometer is an instrument used for detecting and measuring small electrical currents. It operates on the principle that when a current-carrying conductor is placed in a magnetic field, it experiences a force. This force is given by: where:
- is the magnetic field strength,
- is the current passing through the conductor,
- is the length of the conductor in the magnetic field.
In a moving coil galvanometer, a coil of wire is placed in a uniform magnetic field. When a current flows through the coil, it experiences a torque, causing the coil to rotate. The deflection of the coil is proportional to the current flowing through it. A spring is used to restore the coil to its original position, and a scale is used to measure the deflection. Increasing Sensitivity:
The sensitivity of a moving coil galvanometer depends on factors such as:
1. Number of turns in the coil: Increasing the number of turns in the coil increases the torque for a given current, which increases the sensitivity.
2. Area of the coil: Increasing the area of the coil also increases the torque, thus improving the sensitivity.
3. Strength of the magnetic field: A stronger magnetic field increases the force acting on the coil, thus increasing the sensitivity.
4. Spring constant: A lower spring constant makes the galvanometer more sensitive, as it requires less force to produce a given deflection.
By optimizing these factors, the sensitivity of the galvanometer can be increased, allowing it to measure smaller currents more accurately.

Conditions for Interference:
The main conditions for the interference of light are as follows: 1. Coherence: The two light sources must be coherent, meaning they must have a constant phase relationship. 2. Monochromatic light: The light used should be monochromatic (of a single wavelength), as interference depends on the wavelength of the light. 3. Same amplitude: The amplitudes of the two interfering waves should be the same or nearly the same. 4. Superposition principle: The resultant intensity at any point is the sum of the individual intensities due to each wave at that point, following the principle of superposition. Intensity of Light when Path Difference is :
In Young's double-slit experiment, the intensity of light at a point on the screen is given by the equation: where: - is the intensity at a given point,
- is the maximum intensity,
- is the path difference between the two waves,
- is the wavelength of the monochromatic light. For path difference , we substitute into the equation: Since , we have: Thus, the intensity of light at a point where the path difference is is half of the maximum intensity:

Electric Field Outside a Charged Spherical Shell:
According to Gauss' law, the electric field due to a spherically symmetric charge distribution can be determined by considering a Gaussian surface that is a sphere of radius outside the charged spherical shell. The total charge enclosed by the Gaussian surface is the charge on the spherical shell. Gauss's law is expressed as: where: - is the total charge enclosed by the Gaussian surface, and - is the permittivity of free space. Since the electric field is radially symmetric and uniform over the Gaussian surface, the left-hand side of Gauss's law becomes: where is the electric field at the distance from the center of the shell. Therefore, Gauss's law gives: Solving for , we get the formula for the electric field outside the spherical shell: This is the same expression for the electric field as if all the charge were concentrated at the center of the spherical shell. This result holds true outside the shell for any point at a distance from the center.

The difference between conductors and semiconductors can be explained on the basis of their energy band structure, which refers to the ranges of energy that electrons can occupy within a solid. The key differences are:
1. Conductors:
- In conductors, the conduction band and valence band overlap, allowing free movement of electrons. This overlap means that electrons in the valence band can easily move to the conduction band even at low temperatures.
- There is no energy gap between the valence band and the conduction band. As a result, conductors allow the flow of electric current under an applied electric field.
- Example: Metals like copper, silver, and aluminum.
2. Semiconductors:
- In semiconductors, the valence band and the conduction band are separated by a small energy gap called the band gap. Electrons in the valence band need to gain energy to jump to the conduction band. At absolute zero, no electrons are in the conduction band, but at higher temperatures, some electrons gain enough energy to move to the conduction band and conduct electricity.
- The energy gap between the valence band and conduction band is relatively small (around ), which is why semiconductors conduct electricity under certain conditions (e.g., with temperature or doping).
- Example: Silicon, Germanium.

The total electric flux linked with a closed surface is given by Gauss's law:
Where:
- is the total charge enclosed within the surface,
- is the permittivity of free space.
Now, for a dipole, the net charge enclosed is zero because a dipole consists of two equal and opposite charges. Therefore, the total charge enclosed by the surface is zero.
Hence, the total electric flux linked with the closed surface is:
Thus, the correct answer is option (D) zero.

The coefficient of self-inductance is defined as the ratio of the induced emf to the rate of change of current:
The unit of induced emf is volt (V), and the unit of current change rate is ampere per second (A/s). Therefore, the unit of is:
Hence, the correct answer is option (C) henry.

Brewster's Law:
Brewster's law describes the relationship between the angle of incidence, the refractive indices of the two media, and the polarization of reflected light. Brewster's law states that when unpolarized light strikes a transparent dielectric material at a particular angle, called the Brewster angle, the reflected light is completely polarized. At this angle, the reflected and refracted light are perpendicular to each other. The Brewster angle is given by: where: - is the Brewster angle (in degrees or radians),
- is the refractive index of the medium the light is coming from (e.g., air),
- is the refractive index of the medium the light is entering (e.g., glass).
At the Brewster angle, no reflected light is transmitted parallel to the surface, and the reflected light becomes completely polarized. This law is important in the context of optical devices such as polarizing filters.

Refraction of Light from a Spherical Surface:
When light passes from one medium to another at a spherical surface, the relation between the object distance , image distance , and the radius of curvature of the spherical surface is given by the lens-maker's formula. The formula for refraction at a spherical surface is: where:
- and are the refractive indices of the first and second mediums,
- is the object distance,
- is the image distance,
- is the radius of curvature of the surface.
For a thin lens, we have two spherical surfaces with radii and , and the light is refracted twice. The lens-maker's equation for a thin lens in air is: where:
- is the focal length of the lens,
- is the refractive index of the material of the lens,
- is the radius of curvature of the first surface, and
- is the radius of curvature of the second surface.
Focal Length for Concave Lens:
For a concave lens, the radii of curvature are negative ( is negative and is positive). Substituting these values into the lens-maker's equation, we get: Since is negative for a concave lens, the focal length will also be negative. This confirms that the focal length of a concave lens is negative.

To solve this question, we need to apply the mirror equation and the magnification formula. The mirror equation is given by:
Where:
- is the focal length of the concave mirror
- is the image distance (the distance from the mirror to the image)
- is the object distance (the distance from the mirror to the object)
The magnification is given by: Also, the magnification is the ratio of the image height to the object height : Now, the object is placed at a distance of from the mirror. This means the object is placed outside the center of curvature. The image formed when the object is at will be real, inverted, and reduced in size. For a concave mirror, the image formed when the object is at will be at . The image formed at this position is real and smaller than the object. The magnification is: Substituting the values and , we calculate the magnification: So, the magnification is . Since the object height cm, the image height is: Therefore, the length of the image is 4 cm. Hence, the correct answer is option (C) 4 cm.

An electrical capacitor is a device that stores electrical energy in an electric field. It consists of two conductive plates separated by an insulating material (called the dielectric). When a potential difference is applied across the plates, a charge is stored on each plate.
The capacitance of a parallel plate capacitor is given by the expression: Where:
- is the capacitance,
- is the permittivity of free space ( ),
- is the area of one of the plates,
- is the separation between the plates.
The capacitance depends on the following factors:
1. The area of the plates: A larger area increases the capacitance.
2. The separation between the plates: A smaller distance between the plates increases the capacitance.
3. The dielectric material between the plates: The dielectric constant ( ) of the material increases the capacitance. The capacitance with a dielectric is given by: Where is the dielectric constant of the material.

The Lens Maker's formula relates the focal length of a lens to the refractive index of the lens material and the radii of curvature and of the two curved surfaces of the lens.
For a thin lens, the Lens Maker's formula is given by: where:
- is the focal length of the lens,
- is the refractive index of the material of the lens,
- is the radius of curvature of the first surface (convex or concave),
- is the radius of curvature of the second surface (convex or concave).
Effect of Refractive Index ( ):
- As the refractive index increases, the focal length of the lens decreases. A higher refractive index material bends light more sharply, thus focusing light at a shorter distance.
Effect of Radii of Curvature ( and ):
- The radii of curvature of the lens surfaces also affect the focal length. For a convex surface, is positive, and for a concave surface, is negative. The curvature of the surfaces influences the converging or diverging behavior of the lens.

Drift Velocity and Mobility of Electrons:
- Drift velocity ( ) is the average velocity of electrons in a conducting material under the influence of an electric field.
- Mobility ( ) refers to the speed of an electron in a material when exposed to an electric field. It is given by the relation:
Where is the drift velocity, is the mobility, and is the electric field.
For a conductor, the drift velocity can also be related to the current, electron density, and cross-sectional area.
Step 1: Calculate the Electric Field
The electric field in the conductor is given by:
Where is the potential difference and is the length of the conductor.
Step 2: Apply Ohm’s Law
The current in the conductor is related to the resistivity ( ), the current density ( ), and the electric field ( ) by:
Where is the electrical conductivity and is the resistivity of the material. The current density is also related to the drift velocity and the electron density by:
Where: - is the electron density ( ), - is the electron charge ( ), - is the drift velocity.
Equating both expressions for :
Step 3: Calculate the Drift Velocity
Rearranging the equation to solve for :
Substitute the known values: - , - , - , - -m.
Thus, the drift velocity of the electrons is:

Step 1: Definition of Polarisation of Light.
Polarisation of light refers to the orientation of the oscillations of the electric field vector in a particular direction. For natural or unpolarised light, the electric field oscillates in many directions perpendicular to the direction of propagation. However, when light is polarised, its electric field vibrates in a single direction.
Step 2: Mechanism of Polarisation.
- Polarised light is typically obtained from unpolarised light by passing it through a polarising filter or using other methods like reflection, refraction, or scattering.
- The electric field of polarised light oscillates only in one plane (or direction). This can be demonstrated using a Polaroid filter.
Step 3: Diagram showing Polarised and Unpolarised Light.
Step 4: Explanation of Diagram.
- Unpolarised light: The electric field oscillates in all directions perpendicular to the direction of propagation. In the diagram, the electric field vector is randomly oriented in different directions.
- Polarised light: The electric field is oriented in a specific direction, as shown by the uniform vector direction in the diagram.
Final Answer: Polarisation is the process by which the oscillations of the electric field in light are restricted to one plane. The difference is that unpolarised light has electric field oscillations in multiple directions, whereas polarised light has oscillations in only one direction.

Step 1 (What a galvanometer does). A moving–coil galvanometer is a current detector with internal resistance and full–scale current . By itself it measures small currents, not voltage.
Step 2 (Convert to voltmeter). To measure a potential difference without disturbing the circuit, the instrument must draw very little current. We therefore add a large series (multiplier) resistance : .
Choosing so that at full–scale makes a voltmeter whose internal resistance is very large, hence negligible loading of the circuit.
Step 3 (Why others are wrong). A low parallel resistance (shunt) is for ammeters, not voltmeters; a high parallel resistance diverts current unpredictably; a low series resistance draws too much current and loads the circuit.
We must connect a high resistance in series.

Work function with . Using : .
Photon energy at : .
Maximum kinetic energy: .