Magnetic Effects Of Current And Magnetism

To solve the problem, we need to determine the mass ratio of two particles and that have equal charges and are accelerated through the same potential difference. They enter a uniform magnetic field, resulting in circular paths with radii (for ) and (for ).

The formula for the radius of the circular path described by a charged particle in a magnetic field is given by:

Where:

• is the mass of the particle.
• is the velocity of the particle.
• is the charge of the particle.
• is the magnetic field strength.

The kinetic energy of a particle accelerated through a potential difference is:

From the equation of kinetic energy, we find the velocity:

Substituting this velocity back into the expression for :

Simplifying, we get:

From this equation, for particles and , whose radii are and respectively, and using the fact that and are the same for both particles, we have:

Equating the expressions for and :

To find the mass ratio , we divide these two equations:

Hence, the mass ratio is given by:

Thus, the correct answer is:

To find the force acting on the charge, we use the equation for the magnetic force on a moving charge: . Here, , , and .

The cross product can be determined using the determinant method:

Calculating the determinant, we find:

.

Substituting back, the force is:

.

Thus, the value of is 32, which confirms it fits within the given range [32,32].

The problem asks for the magnitude of the net magnetic induction at the common center 'O' of two identical, perpendicular, current-carrying circular loops.

Concept Used:

The solution is based on two key principles of magnetism:

1. Magnetic Field at the Center of a Circular Loop: The magnitude of the magnetic field (magnetic induction) at the center of a circular loop of radius 'a' carrying a current 'I' is given by the Biot-Savart law as:

The direction of this magnetic field is perpendicular to the plane of the loop and is determined by the Right-Hand Thumb Rule.

2. Principle of Superposition: The net magnetic field at any point due to a system of multiple current-carrying conductors is the vector sum of the magnetic fields produced by each conductor individually at that point.

Step-by-Step Solution:

Step 1: Determine the magnetic field due to loop A.

Loop A is a horizontal circular loop. The current 'I' is in the anti-clockwise direction when viewed from above. According to the Right-Hand Thumb Rule (curling the fingers in the direction of the current), the thumb points upwards. Therefore, the magnetic field produced by loop A at the center O is directed vertically upwards. Let's consider the vertical direction to be along the z-axis.

The magnitude of this field is:

So, the vector representation is .

Step 2: Determine the magnetic field due to loop B.

Loop B is a vertical circular loop, perpendicular to loop A. The current 'I' is also anti-clockwise as shown. Let's assume loop B lies in the y-z plane. An anti-clockwise current in this plane (as seen from the positive x-axis) would produce a magnetic field along the positive x-axis, according to the Right-Hand Thumb Rule. Thus, the magnetic field produced by loop B at the center O is directed horizontally.

The magnitude of this field is the same as that for loop A since the radius and current are identical:

So, the vector representation is .

Step 3: Calculate the net magnetic field at the center O.

The net magnetic field is the vector sum of the individual fields:

Since is along the z-axis and is along the x-axis, the two magnetic field vectors are perpendicular to each other.

Final Computation & Result:

Step 4: Find the magnitude of the net magnetic field.

The magnitude of the resultant of two perpendicular vectors is found using the Pythagorean theorem:

Substituting the magnitudes:

The magnitude of the magnetic induction at the centre is or equivalently .

In this problem, we need to analyze the interactions of a proton moving through a region of space with electric and magnetic fields. We are given that the proton moves with constant velocity, indicating that the net force on it is zero.

The forces experienced by a charged particle (like a proton) moving in electric and magnetic fields are given by the Lorentz force law:

where:

• is the total force on the particle,
• is the charge of the particle (positive for a proton),
• is the electric field,
• is the velocity of the particle,
• is the magnetic field.

For the proton to continue moving with constant velocity, the net force must be zero:

We will now analyze each given scenario:

1. Case (A):
   • Here, both fields are zero, so there is no force, allowing the proton to move with constant velocity. This is possible, so (A) is correct.
2. Case (B):
   • If the electric field is zero, to have no net force, the cross product term must also be zero.
   • This is possible if is parallel to or is zero itself.
   • Therefore, (B) is also correct.
3. Case (C):
   • With , the magnetic force is zero, and only the electric field contributes to the force.
   • The presence of an electric field will exert a force, disturbing the constant velocity condition unless .
   • So, (C) is not correct.
4. Case (D):
   • To achieve zero net force, the electric and magnetic forces must cancel each other, i.e., .
   • This is possible with specific orientations of and .
   • Thus, (D) can be correct.

Based on this analysis, the options where the proton can move with constant velocity are (A), (B), and (D).

Given:

- Original least count = - The pitch is increased by 75% (new pitch = ) - The number of divisions on the circular scale is reduced by 50% (new divisions = of the original divisions)

Step 1: Formula for Least Count

The least count of a screw gauge is given by:

Step 2: Apply the Changes

The new pitch is , and the new number of divisions is . Hence, the new least count is:

Step 3: Final Calculation

Substituting , we get:

Final Answer:

The new least count is .

To solve this problem, we need to determine the focal lengths of two plano-convex lenses made of glass with a refractive index . We will use the lens maker's formula to find the focal lengths.

Lens Maker's Formula:

The lens maker's formula is given by:

where is the focal length, is the refractive index of the lens material, is the radius of curvature of the first surface, and is the radius of curvature of the second surface. For a plano-convex lens, , so the formula simplifies to:

Focal Length of the First Lens ( ):

The radius of curvature and the lens is in air, so . Using the simplified lens maker's formula:

Focal Length of the Second Lens ( ):

The radius of curvature and the lens is immersed in a liquid with refractive index 1.2, so the effective refractive index .

Ratio of and :

Finally, the ratio of the focal lengths is given by:

Thus, the ratio of to is .

The correct answer is therefore:

This problem involves a thin plano-convex lens made of glass with a refractive index and immersed in a liquid of refractive index . We need to find the radius of curvature of the lens's curved surface. When the plane side of the lens is silver-coated, the lens acts as a concave mirror with a focal length of 0.2 m. Let's solve it step-by-step.

Step 1: Understand the Concept

When the plane side of the lens is silvered, the system becomes like a mirror-lens system, where the lens and the mirror are in combination. The light will pass through the lens, reflect off the silvered surface, and then pass through the lens again. In such a silvered lens, the effective focal length ( ) is given by:

Given that the entire system behaves like a concave mirror of focal length 0.2 m, we have:

(Negative sign because it's a concave mirror)

Step 2: Using Lens and Mirror Formulas

The focal length of a lens in a medium is given by the lens maker's formula:

For a plano-convex lens, (since the plane surface has infinite radius of curvature), and . Therefore, the formula becomes:

Since the plane surface is silvered, it behaves as a mirror with for a concave surface (since plane surface acts as planar mirror after silvering).

Step 3: Solve Using Given

For a silvered lens:

After finding using lens formula, substitute into the combined formula:

Rearrange to find that:

Equating the expanded expression and solving for , we find:

Conclusion: The radius of curvature of the curved surface of the lens is 0.10 m. This matches with option 2.

To determine the coefficient of friction for a car moving on a banked road, we need to analyze the forces acting on the car and use the given conditions.

The forces involved are:

• Centripetal force required to keep the car moving in a circle of radius .
• The gravitational force acting downward.
• The normal force perpendicular to the surface of the road.
• The frictional force, which can act either up or down the slope depending on the speed of the car relative to the design speed.

Let's consider the car at the verge of slipping, where friction is maximized. The centripetal force required is given by:

Let's resolve the forces parallel and perpendicular to the inclined plane.

Balance of forces perpendicular to the incline gives:

Balance of forces parallel to the incline (where friction opposes slip):

Solving these equations will yield the expression for the coefficient of friction .

First, express from the perpendicular balance:

Substitute in the parallel balance equation:

Simplifying and solving for , you arrive at:

This matches the correct answer:

To solve the given problem, we need to evaluate each of the statements provided and determine their validity regarding solar cells, photodiodes, and LEDs. Let's analyze each statement one by one:

1. Statement A: "The junction area of a solar cell is made very narrow compared to a photodiode."

This statement is incorrect. The junction area of a solar cell is usually larger to absorb more sunlight and generate more current, whereas in a photodiode, the focus is on speed and sensitivity, often involving a more optimized design for specific applications.

1. Statement B: "Solar cells are not connected with any external bias."
2. Statement C: "LED is made of lightly doped p-n junction."

This statement is incorrect. LEDs are typically made with heavily doped p-n junctions to facilitate electron-hole recombination, which is necessary for efficient light emission.

1. Statement D: "Increase of forward current results in a continuous increase in LED light intensity."

This statement can be seen as generally true, but there are limitations at maximum current ratings where further increases can damage the LED or cause efficiency droop (a decrease in efficiency with increased current). However, in typical operating ranges, this statement holds true.

1. Statement E: "LEDs have to be connected in forward bias for emission of light."

This statement is correct. LEDs need to be forward-biased to allow charge carriers to recombine and emit light.

Based on the analysis above, the correct statements are B and E. Therefore, the correct answer is: B, E Only.

To solve this problem, we apply the concept of energy level transitions in the Bohr model of the atom. When an electron transitions between energy levels, it emits or absorbs radiation. The wavelength of the radiation is related to the energy difference between the initial and final states by the equation:

where:

• is the energy difference between the two states.
• is Planck's constant.
• is the speed of light.
• is the wavelength of the radiation emitted or absorbed.

Given:

• Wavelength of radiation emitted from state A to state C:
• Wavelength of radiation emitted from state B to state C:

We need to find the wavelength of the radiation emitted during the transition from state A to state B, denoted as .

To find this, we use the relationship between these wavelengths implied by Bohr's model:

• The energy difference between states is additive, as these are quantized transitions: .

Now, substituting the known wavelengths into the energy relations gives:

Thus, the energy difference for the transition from A to B is:

Combining these terms gives us:

Finding a common denominator:

Thus, , which means the correct option is 3000 Å.