Electromagnetic Induction

Faraday's Law: Given: - Radius
-

Only the component of along the area vector contributes to flux. From direction: Area: EMF:

Flux linkage where is the inductance and is the current.
Therefore,
So, inductance is defined as the ratio of magnetic flux linkage to the current passing through the coil.

When an inductor is connected to an AC source, the voltage across the inductor leads the current through it by ( radians).
Let the instantaneous current through the inductor be , where is the peak current and is the angular frequency of the AC source.
The frequency is related to the angular frequency by .
Given Hz, so rad/s.
The instantaneous voltage across the inductor is .
The instantaneous power developed in the circuit is the product of the instantaneous voltage and the instantaneous current: $ , we can write .
\) .
Since , the frequency of the instantaneous power is: \) Hz, the frequency of the instantaneous power is: \) $ The frequency of the instantaneous power developed in the circuit is 100 Hz.

By Faraday’s law:
Where is the number of turns and is the magnetic flux.

Self-inductance is a property of a coil that opposes any change in the current, and it is analogous to inertia in mechanics. Thus, the correct answer is option (1).

The average power of an electromagnetic wave is related to the electric field by the formula: Where is the power and is the rms value of the electric field. Given the power and the distance , we can use the relationship for the electric field in an electromagnetic wave to find the maximum value of the electric field at this distance. From the formula, we calculate the value of , which corresponds to . Thus, the maximum value of the rms value of the electric field is .

The process of a changing current in a coil inducing an emf in a nearby coil is called mutual induction.
Thus, the correct answer is mutual induction.

We use Faraday’s law of electromagnetic induction: The coil rotates through , causing a change in magnetic flux of per turn. The area of the coil is: Substituting values: Solving: Hence, the number of turns is 504.

When a conductor is placed in a changing magnetic field, circular electric currents called eddy currents are induced in the body of the conductor. These currents circulate within the conductor and dissipate energy in the form of heat.
This principle is used in devices like:
- Induction cooktops
- Magnetic brakes
- Eddy current dampers

An alternating current (ac) generator works on the principle of electromagnetic induction. It converts mechanical energy into electrical energy. The mechanical energy comes from a rotating coil in the generator that moves within a magnetic field, and as the coil rotates, it induces an electromotive force (emf), thereby generating electrical energy. Thus, the correct answer is .

The induced electromotive force (emf) in an inductor is given by Faraday's law of induction, which states that the induced emf is proportional to the rate of change of current through the inductor: $ is the inductance of the inductor and is the rate of change of current.
The negative sign indicates the direction of the induced emf (Lenz's law), but we are interested in the magnitude here.
Given: Initial current A Final current A Time interval s Induced emf V The change in current is A.
The rate of change of current is .
Now, we can find the inductance using the magnitude of the induced emf: \) $ The unit of inductance is Henry (H), and 1 H = 1 VsA .
Therefore, the inductance of the inductor is H.

Lenz's law, in electromagnetism, states that an induced electric current flows in a direction such that the current opposes the change that induced it.

When two coaxial coils carrying the same current in the same direction are brought near each other, their magnetic fields interact. According to Lenz's Law, the system will adjust to oppose this change. The mutual inductance between the coils increases due to their proximity, which affects the total magnetic field.

The specific interaction causes an induced electromotive force (EMF) in each coil. Due to Lenz's Law, this EMF opposes the original current. Thus, the overall effect is a reduction in the net current flowing in both coils.

Hence, when two coaxial coils with identical currents are brought close together, the current in both coils reduces. Therefore, the correct answer is: Decreases.

Step 1: Induced EMF in a Rotating Wheel The induced emf between the axle and rim is given by: where: - = Magnetic field strength, - = Length of the spoke, - = Angular velocity. Step 2: Comparing Initial and Final Cases Since number of spokes does not affect the induced emf, we compare the effect of rotation speed. Initial conditions: where rev/min. Final conditions: Since rev/min, Conclusion Thus, the correct answer is:

Energy stored in an inductor: .
Given: mH = H, A.
J.

Step 1: Impedance Formula for a Series RLC Circuit The impedance is given by: where: - (resistance), - (capacitive reactance), - (inductive reactance). Step 2: Computing Net Reactance Step 3: Calculating Impedance Conclusion Thus, the correct answer is:

Step 1: Calculate useful power for visible radiation
Step 2: Use formula for intensity
Given

Step 1: Use Faraday's law of electromagnetic induction.
Induced emf: Step 2: Insert given values.
Step 3: Calculate emf. Step 4: Select the correct option.
The induced emf is approximately 0.72 mV, which matches option (3).

Step 1: Formula for resonant frequency
Step 2: Effect of dielectric
Inserting a dielectric increases capacitance by factor : Step 3: New resonant frequency
Step 4: Conclusion
New resonant frequency is .

Radius of the disc m.
Angular speed .
Magnetic field strength .
The plane of the disc is perpendicular to the magnetic field, so the field lines are parallel to the axis of rotation.
The emf induced between the centre and a point on the rim of a rotating metallic disc in a uniform magnetic field perpendicular to its plane is given by: Substitute the given values: .
This matches option (4).

In an electromagnetic wave, the electric and magnetic field vectors are always perpendicular to each other and to the direction of propagation of the wave. They are also in phase, meaning they reach their maximum and minimum values at the same time.

Step 1: Matching load using transformer turns ratio
Impedance matching condition for transformers: where , Step 2: Taking square root on both sides
Step 3: Final Answer
Hence, the required ratio of primary to secondary turns is

Step 1: Use the energy density formula
Energy density (energy per unit volume) in a magnetic field is given by: where is permeability of free space.
Step 2: Express in terms of and
Step 3: Substitute and simplify
Step 4: Conclusion
Energy stored per unit volume is .

Net voltage:

Faraday's law states that the induced electromotive force (emf) in a coil is equal to the negative rate of change of magnetic flux linked with it. Mathematically, $ \phi = 50t^2 + 4 t = 2 \, \text{s} \text{emf} = -100(2) = -200 \, \text{V} V = IR V I R $

The emf induced in the coil is given by: $ L \frac{dI}{dt} L$ to be 4 mH.

Resistance .
Frequency .
Power factor .
The circuit is an LR series circuit.
The power factor is given by , where is the impedance of the circuit.
, where is the inductive reactance.
Given : Now use : Alternatively, if , then .
Also, .
For an LR circuit, .
The inductance is 4 H.
This matches option (2).

Step 1: Induced EMF in a Rotating Wheel The induced emf between the axle and rim is given by: where: - = Magnetic field strength, - = Length of the spoke, - = Angular velocity. Step 2: Comparing Initial and Final Cases Since number of spokes does not affect the induced emf, we compare the effect of rotation speed. Initial conditions: where rev/min. Final conditions: Since rev/min, Conclusion Thus, the correct answer is:

Step 1: Impedance Formula for a Series RLC Circuit The impedance is given by: where: - (resistance), - (capacitive reactance), - (inductive reactance). Step 2: Computing Net Reactance Step 3: Calculating Impedance Conclusion Thus, the correct answer is:

Radius of the disc m.
Angular speed .
Magnetic field strength .
The plane of the disc is perpendicular to the magnetic field, so the field lines are parallel to the axis of rotation.
The emf induced between the centre and a point on the rim of a rotating metallic disc in a uniform magnetic field perpendicular to its plane is given by: Substitute the given values: .
This matches option (4).

Resistance .
Frequency .
Power factor .
The circuit is an LR series circuit.
The power factor is given by , where is the impedance of the circuit.
, where is the inductive reactance.
Given : Now use : Alternatively, if , then .
Also, .
For an LR circuit, .
The inductance is 4 H.
This matches option (2).

Step 1: Identify the Formula for Induced EMF in Terms of Self-Inductance
When the current in a coil changes, an electromotive force (emf) is induced in the coil itself. This phenomenon is called self-induction. The induced emf ( ) is directly proportional to the rate of change of current ( ). The constant of proportionality is the self-inductance ( ) of the coil. The formula for the induced emf is: The negative sign indicates that the induced emf opposes the change in current (Lenz's Law). When calculating the magnitude of the self-inductance, we often use the magnitude of the emf and the rate of change of current: Step 2: Extract Given Values from the Problem
Given: \begin{itemize} \item Initial current, \item Final current, \item Change in time, \item Average induced emf, \end{itemize} Step 3: Calculate the Rate of Change of Current ( )
The change in current, . The rate of change of current is: The magnitude of the rate of change of current is . Step 4: Compute the Self-Inductance ( )
Using the magnitude form of the induced emf formula: Rearrange the formula to solve for : Substitute the values: Step 5: Analyze Options
\begin{itemize} \item Option (1): 20 H. Incorrect. \item Option (2): 4 H. Correct, as it matches our calculated self-inductance. \item Option (3): 2 H. Incorrect. \item Option (4): 40 H. Incorrect. \end{itemize}