Electrostatic Potential

The potentiometer works on the principle that the potential difference across a uniform wire is directly proportional to the length of the wire. The formula to calculate the emf of the secondary cell is: Where: - is the emf of the secondary cell. - is the emf of the primary battery. - is the balancing length. - is the total length of the potentiometer wire. Substituting the values into the formula: Solving for : Thus, the emf of the secondary cell is 4 V. Therefore, the correct answer is (A) 4 V.

The problem involves the Doppler Effect, which describes the change in frequency of a wave in relation to an observer who is moving relative to the wave source.

When the source (train) is moving towards the observer, the sound waves are compressed, which leads to an increase in the frequency heard by the observer. When the train passes the observer and starts moving away, the sound waves are stretched, and the frequency decreases.

- As the train approaches the observer, the observed frequency will increase.
- When the train passes the observer, the frequency will suddenly drop and become lower than the initial frequency as the train moves away.

The graph that represents this situation will show:
- An increasing frequency as the train approaches the observer.
- A sharp drop in frequency once the train moves past the observer.

This is accurately represented by option (D), where the frequency increases as the train gets closer and drops immediately after it passes.

Given:
Velocity of proton:
Electric field:
Magnetic field:
Charge of proton:

Net force on the proton:

First calculate :

Using cross product:



Now compute :


Add them:


Now compute force:


Now find the magnitude:


Approximate net force:
This is closest to:

Consider the following charges placed at the corners of a square:

• +q at corner A
• +2q at corner B
• +q at corner C
• -2q at corner D

We need to find the net force on the unit positive charge placed at the center of the square (O).

Step 1: Coulomb's Law

The force between two charges is given by Coulomb's law:

Step 2: Analyzing the forces exerted by each charge on the unit positive charge at O

• Force exerted by +q at A: The force will be attractive and directed along the line AO, from A to O.
• Force exerted by +2q at B: The force will be attractive and directed along the line BO, from B to O.
• Force exerted by +q at C: The force will be attractive and directed along the line CO, from C to O.
• Force exerted by -2q at D: The force will be repulsive and directed along the line DO, from D to O.

Step 3: Direction of the forces:

Since the charges at A, B, and C are positive, they attract the unit positive charge at O. The charge at D is negative and repels the unit positive charge. Therefore, the forces along the lines AO, BO, and CO have equal magnitudes but are mutually perpendicular, directed towards the center of the square.

The force along the line DO is repulsive and directed away from the charge at D, and it has a different magnitude from the forces along AO, BO, and CO.

Step 4: Vector Addition of Forces

The forces along AO, BO, and CO are mutually perpendicular. Since they have equal magnitudes and are directed at 90° to each other, they will cancel out each other when added vectorially.

The force along the line DO will not cancel out completely. It will have a net component along the diagonal BD of the square. The net force on the unit positive charge at O is along the diagonal BD.

Therefore, the net force on the unit positive charge at the center O is along the diagonal BD (option D).

The self-inductance of a solenoid can be calculated using the formula:

Given:

• turns

Using the formula, we substitute the given values:

Therefore, the self-inductance of the solenoid is 1.0 Henry. Hence, the correct option is (C) 1.0 Henry.

Given:

Step 1: Square the expression

Step 2: Average over one time period

The average of and over one time period is .

The average of over one time period is 0.

Therefore, the average of over one time period is:

Step 3: Take the square root

Comparing this result to the given options:

• (1) – This matches our calculated expression.
• (2) – This is not equivalent to our expression.
• (3) – This is not equivalent to our expression.
• (4) – This is not equivalent to our expression.

Therefore, the correct answer is: (A) .

According to the Biot-Savart Law, the magnetic field at a point due to a current-carrying loop is given by the equation:

Where:

• B is the magnetic field (induction)
• is the permeability of free space (constant)
• n is the number of turns in the coil
• I is the current flowing through the coil
• R is the radius of the coil
• x is the distance between the point on the axis and the center of the coil

In this case, the point on the axis is a distance of from the center of the coil. Plugging this value into the equation, we have:

Simplifying the denominator:

Therefore, the correct option is (1) .

The drift velocity of a charged particle in an electric field is related to its mobility and the electric field strength by the following equation:

Given:

• Electric field,
• Mobility,

Substituting the given values into the equation:

Therefore, the drift velocity is . However, none of the given choices exactly match this result, there must be a typo in the electric field value. Let's assume the electric field is . Then:

The correct answer is

Given two charges and , located 40 cm (0.4 m) apart, calculate the electric potential at the midpoint between the charges. The midpoint is at a distance of 20 cm (0.2 m) from each charge.

Step 1: Formula for Electric Potential

The electric potential at a point due to a charge is given by the formula:

Where:

• k: Coulomb's constant
• q: The charge producing the potential
• r: The distance from the charge to the point where the potential is being calculated

Step 2: Calculate the potential at the midpoint due to each charge:

Potential due to (V1):

Where is the distance from to the midpoint. Substituting the given values:

Potential due to (V2):

Where is the distance from to the midpoint. Substituting the given values:

Step 3: Calculate the total potential at the midpoint:

The total potential is the sum of the potentials due to and :

Substituting the values for and :

Therefore, the potential at the midpoint between the two charges is approximately: . The closest value is (D) .

The torque acting on an electric dipole in a uniform electric field is given by the formula:

Given:

• Dipole moment (p):
• Electric field (E):
• Angle (θ):

Step 1: Substitute the given values into the formula:

Step 2: Use the value of :

Step 3: Simplify the expression:

Therefore, the magnitude of the torque acting on the dipole is: . The correct answer is (B) .

Step 1: Equating Gravitational Force and Drag Force

The gravitational force acting on the drop is given by:

The drag force due to viscosity is given by Stokes' law:

At terminal velocity, these two forces are balanced:

Step 2: Solving for Mass

Rearranging the equation for :

The mass of the drop can also be expressed in terms of its volume and density:

Substitute this expression for into the equation:

Step 3: Solving for the Radius

Rearrange the equation to solve for :

Step 4: Substituting the Given Values

Given values:

• Viscosity
• Velocity
• Density
• Gravitational acceleration

Substitute these values into the equation:

Simplifying:

Taking the square root of both sides:

Step 5: Finding the Charge

The electric force acting on the oil drop is given by:

The electric field strength is given as:

The gravitational force can also be expressed as:

Equating the two expressions for the electric force:

Solving for :

Substitute the known values:

Simplifying:

Therefore, the magnitude of the charge is: (option D).

The electromagnetic spectrum encompasses a wide range of radiation types, each characterized by its wavelength and frequency. The relationship between wavelength ( ) and frequency ( ) is given by , where is the speed of light. This means that radiation with a higher wavelength has a lower frequency, and vice versa.

Considering the common types of electromagnetic radiation:

• Radio waves
• Microwaves
• Infrared radiation
• Visible light
• Ultraviolet radiation
• X-rays
• Gamma rays

The order of these radiations from longest to shortest wavelength is:

1. Radio waves
2. Microwaves
3. Infrared
4. Visible Light
5. Ultraviolet
6. X-Rays
7. Gamma Rays

Therefore, among the options typically presented, microwaves have a higher wavelength than other radiation types such as infrared, visible light, ultraviolet, X-rays and Gamma rays. Radio waves have the highest wavelength in the entire EM spectrum.

Thus, the electromagnetic radiation with the highest wavelength from the expected choices is:

Microwave

The impedance of a series LCR circuit can be calculated using the formula:

Where is the resistance, is the inductive reactance, and is the capacitive reactance.

Given values:

• R = 300 Ω
• L = 0.9 H
• C = 2.0 µF
• ω = 1000 rad/sec

The inductive reactance can be calculated as follows:

Substituting the given values:

The capacitive reactance can be calculated as follows:

Substituting the given values:

Now, we can substitute the values of , , and into the impedance formula:

Therefore, the impedance of the circuit is 500 Ω. So the correct option is (A) 500 Ω.

The resonance frequency ( ) of an LC circuit is given by the formula:

Substituting the values for the capacitance ( ) and inductance ( ):

Therefore, the resonance frequency of the circuit is: .