Electromagnetic Induction

According to Faraday's law of electromagnetic induction, an emf is induced in a coil if the magnetic flux linked with the coil changes.

In this problem, we need to find the mutual inductance between the two coils. The relationship between the induced emf in the secondary coil and the change in current in the primary coil is given by the formula: Where: - is the induced emf in the secondary coil, - is the change in current through the primary coil, - is the time interval during which the current changes. Now, rearranging the formula to solve for : Substitute the known values: Thus, the mutual inductance of the two coils is . Therefore, the correct answer is Option (D): .

The emf induced in a rotating rod in a magnetic field is given by the formula: where: - is the magnetic field strength, - is the angular velocity of the rod, - is the length of the ro(D) First, let's calculate the angular velocity : - The frequency of rotation is , - The angular velocity is . The length of the rod is , and the magnetic field strength is . Now, substitute these values into the formula for emf: Thus, the induced emf across the ends of the rod is .

We know the dimensional formulas of capacitance and electric potential :
1. Capacitance : The dimensional formula for capacitance is: where is charge and is electric potential. The dimensional formula for charge is , and the dimensional formula for electric potential is .
Thus, the dimensional formula for capacitance is:
2. Electric potential : The dimensional formula for electric potential is already provided as: Now, the expression has dimensions: Simplifying:
Thus, the dimensional formula of is: Therefore, the correct answer is:

To determine the emf induced in the coil, we use Faraday's law of electromagnetic induction, which states that the induced emf ( ) in a coil is equal to the negative rate of change of magnetic flux ( ) through the coil. Mathematically, this is expressed as:

Given the magnetic flux linked with the coil:

we first calculate the derivative of with respect to time :

Applying basic differentiation rules, we get:

Substituting seconds into the derivative, we find:

Hence, the induced emf is:

Since the question asks only for the magnitude of the emf, the answer is 22 V.

The induced e.m.f. is given by the negative rate of change of magnetic flux: Given: Differentiate with respect to time : Now, to find the induced e.m.f. at seconds, substitute :
Thus, the magnitude of the induced e.m.f. is 49 V, but since the question asks for the value in the 3rd second, the correct answer is .

The induced emf in a conducting loop moving in a magnetic field is given by Faraday's Law of Induction: Where: - is the magnetic flux, - is the rate of change of flux. The magnetic flux is given by: Here: - is the magnetic field, - is the radius of the loop. Since the radius is shrinking at a rate of , we can differentiate the flux: Substitute the values: Simplifying this:
Thus, the induced emf is .

The emf induced in a rotating rod placed in a magnetic field is given by the formula: Where: - is the induced emf, - is the magnetic field strength, - is the angular velocity, - is the length of the rod. The angular velocity is related to the frequency by: Now, given: - , - , - , First, calculate the angular velocity: Now substitute the values into the formula for emf: Thus, the emf developed across the ends of the rod is volts.

The induced emf ( ) in the secondary coil is given by Faraday's law of induction: where: - is the number of turns in the coil (250 turns), - is the magnetic flux through a coil, and - is the rate of change of the magnetic flux. The magnetic flux is given by: where: - is the magnetic field due to the solenoid, and - is the area of cross-section of the solenoid. The magnetic field inside the solenoid is given by: where: - is the permeability of free space, - is the number of turns in the solenoid (3000 turns), - is the length of the solenoid (0.2 m), and - is the current passing through the solenoid (2 A). Substitute the given values into the equation for : Now, substitute into the equation for flux : The rate of change of flux is: Finally, the induced emf is: Thus, the induced emf produced is .

The magnetic field at the center of a coil due to a current is given by: where is the permeability of free space, is the number of turns, is the current, and is the radius of the coil. For coil A, the magnetic field at the center is: For coil B, the current is , so the magnetic field at the center is: Now, since the coils are placed concentrically and their planes are perpendicular to each other, the net magnetic induction will be the vector sum of the magnetic fields from each coil. Since the angle between the magnetic fields is 90°, we use the Pythagorean theorem to find the resultant: Substituting the expressions for and : Thus, the net magnetic induction is , which corresponds to option (B).

The coefficient of mutual induction between two coils is defined as the ratio of the flux linkage in one coil to the current in the other coil that produces the flux. It is given by: where: - is the flux linked with coil 2 due to the current in coil 1, - is the current in the first coil. The unit of flux ( ) is the weber (Wb), and the unit of current ( ) is the ampere (A). Thus, the unit of mutual induction is: because one weber is equal to one volt-second, and mutual induction involves the flux linkage, which is measured in volt-seconds per ampere. Thus, the unit of the coefficient of mutual induction is volt second / ampere, which corresponds to option (A).