Electromagnetic Waves

The problem involves a special metal , which conducts electricity without any resistance, placed in a closed wire loop. The loop generates a compensating flux in response to a changing magnetic flux through it, inducing a current that cannot decay due to the zero resistance of the material. This induced current generates a magnetic moment that repels the source of the magnetic field or flux. The system consists of a loop of radius , with the center at the origin, and a magnetic dipole of moment brought along the axis of the loop from infinity to a distance from the center of the loop (with ). The north pole of the dipole always faces the loop.

Step 1: Magnetic Field Due to a Dipole

The magnitude of the magnetic field produced by a magnetic dipole of moment at a point on its axis, at a distance , is given by the formula:

Where is the permeability of free space, and is the magnetic moment of the dipole. This field is along the axis of the dipole, pointing towards the dipole’s south pole if the north pole is facing the loop.

Step 2: Induced Current in the Loop

The wire loop of radius will induce a current to generate a compensating flux in response to the magnetic field produced by the dipole. According to Faraday's Law of Induction, the induced electromotive force (emf) is given by:

Where is the magnetic flux through the loop. The induced current in the loop will be proportional to the emf, and because the loop has zero resistance, the current will be proportional to the magnetic field at the location of the loop. Since the magnetic field is given by , the induced current will be proportional to this field.

Step 3: Current Induced in the Loop

The induced current in the loop is proportional to the magnetic field at a distance from the dipole’s center. Thus, the current induced in the loop will be proportional to:

Final Answer:

The induced current in the loop will be proportional to .

The problem involves two concentric circular loops, one of radius and the other of radius , lying in the -plane with the origin as their common center. The smaller loop carries a current in the anti-clockwise direction, and the larger loop carries a current in the clockwise direction. It is given that . We are tasked with determining the behavior of the magnetic field at a point in the -plane and its dependence on the radial distance from the center of the loops.

Step 1: Magnetic Field Due to a Current-Carrying Loop

The magnetic field at a point due to a current-carrying loop is determined using the Biot-Savart law. For a loop of radius carrying a current , the magnetic field produced at any point depends on the geometry of the loop, the current in the loop, and the location of the point relative to the loop. For a point on the axis of the loop, the magnetic field is directed along the axis, and its magnitude is influenced by both the radius of the loop and the distance from the loop to the point of interest. However, for points in the plane of the loop, the situation becomes more complex, but the magnetic field can still be calculated using the principle of superposition.

Step 2: Symmetry of the Problem

In this specific problem, we are dealing with two concentric loops. The key feature of this setup is the symmetry of the configuration. The two loops lie in the same plane and share a common center. Since the loops are concentric and circular, the magnetic field at any point in the plane can be considered as the sum of the contributions from each loop. Because both loops are centered at the origin, and the system exhibits radial symmetry, the magnetic field at any point in the -plane will depend on the radial distance from the center, but not on the individual coordinates and . This is a consequence of the fact that the system is rotationally symmetric around the origin.

Step 3: Dependence of the Magnetic Field on the Radial Distance

Since both loops are concentric, the magnetic field produced at any point in the plane will have radial symmetry. This means that the magnetic field vector at a given point will point either directly towards or away from the center of the loops, and its magnitude will only depend on the radial distance from the center. The contributions from both loops will add up depending on the distance from each loop, but the resulting magnetic field will only vary with the distance , not with the individual values of or .

Step 4: Resulting Magnetic Field at Any Point

The magnetic field at any point in the -plane due to the two concentric current-carrying loops depends on the radial distance and is the result of the combined contributions from the two loops. The field produced by each loop is symmetric, and thus the net field depends on the radial distance from the center of the loops alone. This means that the magnitude of the magnetic field at any point will vary with but not directly with the specific values of and individually.

Final Answer:

The magnitude of the magnetic field depends on and only through the radial distance . This means that the magnetic field is a function of the distance from the center of the loops and does not depend on the individual coordinates and .

Given:

• Current in wire,
• Semi-circular rod moves with velocity
• Distances of rod ends from wire:
• Capacitor , Resistor
• ,

Step 1: Magnetic field at distance r from a long wire

Step 2: emf induced across the rod due to motion in magnetic field

Since the rod moves perpendicular to the field, the emf is: Substitute values:

Step 3: Maximum charge on capacitor

Maximum charge:

Correct Answer: Option (C): Maximum charge on capacitor is

Step-by-step Solution:

Concept: When charged particles are accelerated through a potential difference , they gain kinetic energy equal to the work done by the electric field:

In a magnetic field perpendicular to the velocity, the radius of circular motion is given by: Hence,

Step 1: Write mass and charge for each particle

• α-particle:
  • Mass
  • Charge
• Singly charged sulfur ion:
  • Mass
  • Charge

Step 2: Take ratio of radii

Final Answer: 4

Step 1: Given Information
The physical quantity is defined as: where: - is the electric field, - is the magnetic field, - is the permeability of free space.
We are asked to determine the dimensions of and match them with the dimensions of one of the given quantities.

Step 2: Determining the Dimensions of
The dimensions of depend on the cross product of the electric field and the magnetic field , and the permeability of free space . We start by analyzing the dimensions of each term: - The dimensions of the electric field are given by: where is mass, is length, is time, and is the electric current. - The dimensions of the magnetic field are: - The permeability of free space has the dimensions: Now, the cross product results in the dimensions: Dividing by , we get the dimensions of :

Step 3: Matching the Dimensions
Now, we compare the dimensions of with the given options. We are asked to find the quantity that has the same dimensions as . The options are: - (B) - (D)
The dimensions of force are: Therefore, the dimensions of are: This matches the dimensions of . Therefore, both options (B) and (D) are correct.

Final Answer:
The correct options are: - (B) - (D)

Step 1: Given Information
Two concentric circular loops are given, one with radius and the other with radius . The loops are in the -plane, sharing the origin as their common center.
- The smaller loop carries current in the anti-clockwise direction.
- The larger loop carries current in the clockwise direction, and we are given that .
- We are asked to determine the magnetic field at a point in the -plane.

Step 2: Magnetic Field Due to Circular Current Loop
The magnetic field at a point due to a current-carrying circular loop is given by the Biot-Savart law. The field produced by each loop at any point in the plane of the loop depends on the distance from the point to the center of the loop.
- For a point in the plane of the loop, the magnetic field produced by each loop is perpendicular to the plane of the loop, pointing either upward or downward, depending on the direction of the current.
- The contribution of each loop to the magnetic field at any point depends on the radial distance from the center.

Step 3: Analyzing Statement A
We are asked to verify whether is perpendicular to the -plane at any point in the plane.
- The magnetic field produced by each loop is directed along the axis perpendicular to the plane of the loop, which is the -axis in this case.
- Therefore, the magnetic field at any point in the -plane will be along the -axis, meaning it will be perpendicular to the -plane.
Hence, statement (A) is correct.

Step 4: Analyzing Statement B
We are asked to verify whether depends on and only through the radial distance .
- From the Biot-Savart law and the symmetry of the problem, the magnetic field at any point in the -plane due to both loops will depend only on the radial distance from the center of the loops.
- Since the magnetic field produced by each loop is a function of the radial distance , and because the loops are symmetric about the origin, the field will only depend on , which is a function of and as .
Hence, statement (B) is correct.

Final Answer:
Both statements (A) and (B) are correct:
- (A) is perpendicular to the -plane at any point in the plane.
- (B) depends on and only through the radial distance .