Practice Questions

An inductor with an inductance of $2.5 \text{ H}$ carries a steady current of $4.0 \text{ A}$. Calculate the magnetic potential energy stored in the inductor.

Compare the mutual inductance of two identical circular coils when they are placed co-axially with their planes parallel versus when they are placed with their planes perpendicular to each other.

Analyze how the peak value of the induced EMF ($\varepsilon_0$) in an AC generator changes if the frequency of rotation is doubled, keeping all other parameters constant.

What is the primary function of an AC generator?

Apply Faraday's law of induction to explain why no current is induced in a closed loop held stationary in a strong, uniform magnetic field.

Critique the following statement: 'In Experiment 6.3, if the tapping key K is pressed and held, a constant, non-zero current will be induced in coil C1 because of the steady magnetic field from coil C2.'

Name the SI unit of magnetic flux and define it.

Propose a scenario where a non-zero EMF is induced in a circuit, but no current flows. Justify your proposal.

Justify the statement that self-inductance is the electromagnetic analogue of mass (inertia) in mechanics.

Define the phenomenon of electromagnetic induction.

Recall the formula for the motional electromotive force (emf) induced in a straight conductor of length $l$ moving with velocity $v$ perpendicular to a uniform magnetic field $B$.

Two coils have a mutual inductance of $1.25 \text{ H}$. The current in the primary coil changes from $2.0 \text{ A}$ to $10.0 \text{ A}$ in $0.20 \text{ s}$. Calculate the magnitude of the average EMF induced in the secondary coil.

Calculate the self-inductance of a long air-cored solenoid having a length of $1.5 \text{ m}$, a cross-sectional area of $25 \text{ cm}^2$, and $500$ turns per meter. Use $\mu_0 = 4\pi \times 10^{-7} \text{ T} \cdot \text{m/A}$.

Evaluate why inserting a soft iron rod into the coils in Faraday's third experiment dramatically increases the galvanometer deflection. Formulate an explanation based on the concepts of magnetic materials and magnetic flux.

Analyze the role of the back EMF in an inductor when a switch is suddenly opened in a DC circuit. Explain why this can result in a spark across the switch contacts.

Explain the concept of mutual inductance. List the factors on which the mutual inductance of a pair of coils depends.

Define self-inductance and state its SI unit.

Critique the experimental setup in Example 6.7, where a wheel with 10 metallic spokes rotates in a magnetic field. Justify why the number of spokes is described as 'immaterial' to the final induced EMF between the axle and the rim.

State and explain Faraday's laws of electromagnetic induction.

Design a practical device that utilizes the principle of self-induction to oppose sudden changes in current. Justify how your design achieves this function, referencing the formula for self-induced EMF.

Explain Lenz's law and describe how it is a consequence of the law of conservation of energy.

How is the induced emf in a coil related to the rate of change of current in the same coil? Name the proportionality constant involved.

An airplane is flying horizontally at a speed of $900 \text{ km/h}$. Solve for the potential difference developed between the ends of its wings having a span of $20 \text{ m}$, if the vertical component of the Earth's magnetic field is $5.0 \times 10^{-5} \text{ T}$.

Describe the observation in the galvanometer when, in Experiment 6.1 (bar magnet and coil): (a) the North pole of the magnet is moved quickly towards the coil, (b) the magnet is held stationary inside the coil, and (c) the South pole of the magnet is moved away from the coil.

Calculate the magnetic flux through a rectangular coil of dimensions $12 \text{ cm} \times 5 \text{ cm}$ placed in a uniform magnetic field of $0.25 \text{ T}$. The plane of the coil is inclined at an angle of $60^{\circ}$ to the direction of the magnetic field.

Analyze the motion of a strong bar magnet dropped vertically through a horizontal copper ring. Compare this motion to the magnet being dropped in the absence of the ring.

An AC generator consists of a coil of $50$ turns and area $0.25 \text{ m}^2$ rotating at an angular speed of $60 \text{ rad/s}$ in a uniform magnetic field of $0.30 \text{ T}$. Calculate the maximum EMF generated in the coil. Also, analyze the effect on the maximum EMF if the angular speed is doubled.

Design an experiment to prove that the induced EMF in a rotating coil (AC generator) is sinusoidal in nature, as predicted by the equation $\varepsilon = \varepsilon_0 \sin(\omega t)$.

Evaluate the energy storage capabilities of an inductor versus a capacitor. Formulate the expressions for energy density for both and critique their dependence on electric and magnetic fields respectively.

Design an experiment to demonstrate mutual inductance between two co-axial solenoids. Propose three distinct methods to increase the magnitude of the induced EMF in the secondary coil, justifying each method based on the principles of electromagnetic induction.

Justify Lenz's law as a direct consequence of the law of conservation of energy. Use the example of pushing the North pole of a bar magnet towards a closed conducting loop to support your argument.

Describe the three key experiments performed by Faraday and Henry that demonstrated the phenomenon of electromagnetic induction. Summarize the main conclusion drawn from these experiments.

Propose a reason why the core of a transformer or the armature of a generator is made of laminated sheets of soft iron instead of a solid block. Justify your reasoning based on the principles of electromagnetic induction.

A student claims that since motional EMF ($\varepsilon = Blv$) is produced by a magnetic force, and magnetic forces do no work, the generation of electrical energy in this way violates the work-energy theorem. Critique this argument.

A long solenoid has $1000$ turns per meter and a cross-sectional area of $5 \text{ cm}^2$. A steady current of $2 \text{ A}$ flows through it. Recall the necessary formulas and calculate the magnetic flux linked with each turn inside the solenoid. (Use $\mu_0 = 4\pi \times 10^{-7} \text{ T m/A}$)

A metallic disc of radius $R = 15 \text{ cm}$ is rotated about its central axis with a constant angular velocity $\omega = 50 \text{ rad/s}$. A uniform magnetic field of magnitude $B = 0.20 \text{ T}$ is applied perpendicular to the plane of the disc. Calculate the induced EMF between the center and the rim of the disc.

Contrast the physical mechanism that generates the induced EMF in a conductor moving through a constant magnetic field (motional EMF) with that of a stationary conductor in a time-varying magnetic field.

Demonstrate with a suitable example how Lenz's law is a direct consequence of the law of conservation of energy. Consider the case of pushing a bar magnet towards a closed conducting loop.

A square loop of side $20 \text{ cm}$ and resistance $1.0 \Omega$ is placed in a magnetic field directed into the page. The field's magnitude is decreasing with time according to the equation $B(t) = (2.0 - 0.5t) \text{ T}$, where $t$ is in seconds. Formulate an expression for the induced current in the loop as a function of time and justify its direction.

Propose a method to determine the direction of induced current in a loop without using Lenz's Law, but instead by applying the Lorentz force on charge carriers. Justify your proposed method using the scenario from Figure 6.10.

Explain the principle, construction, and working of a simple AC generator. Recall the expression for the instantaneous emf produced.

Recall the expression for the magnetic energy stored in an inductor. Using this and the formula for the self-inductance of a long solenoid, $L = \mu_0 n^2 A l$, explain the steps to derive an expression for the magnetic energy density ($u_B$) in terms of the magnetic field $B$.

A circular coil with $200$ turns and a radius of $5.0 \text{ cm}$ is placed in a magnetic field directed perpendicular to its plane. The magnitude of the magnetic field varies with time according to the equation $B(t) = 0.02t + 0.01t^2$, where $B$ is in Tesla and $t$ is in seconds. Calculate the magnitude of the induced EMF in the coil at $t = 5.0 \text{ s}$.

Evaluate the generation of motional EMF in a rectangular loop PQRS where arm PQ of length $l$ is moved with velocity $v$ in a uniform magnetic field $B$. Formulate the expression for the induced EMF using both Faraday's law of induction and the Lorentz force on charge carriers, and justify that both approaches yield the same result, $\varepsilon = Blv$.

A square loop of side $20 \text{ cm}$ is placed perpendicular to a uniform magnetic field of $0.5 \text{ T}$. The magnetic field is decreased to zero in $0.2 \text{ s}$. Recall the formula for magnetic flux and summarize the steps to find the magnitude of the induced emf.