1. ⇒ (MHT CET 2023 11th May Evening Shift )

The magnetic flux through a circuit of resistance ' $R$ ' changes by an amount $Δ ϕ$ in the time $Δ t$ . The total quantity of electric charge ' $Q$ ' which passes during this time through any point of the circuit is

A. $- \frac{Δ ϕ}{Δ t} + R$

B. $\frac{Δ ϕ}{R}$

C. $\frac{Δ ϕ}{Δ t}$

D. $\frac{Δ ϕ}{Δ t} \times R$

Correct Option is (B)

According to Faraday's law of electromagnetic induction,

$\begin{aligned} ε = \frac{Δ ϕ}{Δ t} \\ IR = \frac{Δ ϕ}{Δ t} \\ I = \frac{Δ ϕ}{Δ t \times R} \\ I \times Δ t = \frac{Δ ϕ}{R} \end{aligned}$

$∴$ The total quantity of electric charge passing through the circuit is

$Q = \frac{Δ ϕ}{R}$

2. ⇒ (MHT CET 2023 11th May Evening Shift )

A coil having effective area ' $A$ ' is held with its plane normal to a magnitude field of induction ' $B$ '. The magnetic induction is quickly reduced to $25 %$ of its initial value in 1 second. The e.m.f. induced in the coil (in volt) will be

A. $\frac{BA}{4}$

B. $\frac{BA}{2}$

C. $\frac{3 BA}{8}$

D. $\frac{3 BA}{4}$

Correct Option is (D)

The formula for induced emf is $e = \frac{Δ ϕ}{Δ t}$ , where $ϕ = BA$

Here, the area is constant and the magnetic field is changing.

$\begin{aligned} ∴ & Δ ϕ & = Δ BA \\ ∴ & Δ ϕ & = A \cdot Δ B \\ ∴ & Δ B & = B_{1} - B_{2} \end{aligned}$

$B_{1} = B and B_{2} = \frac{25}{200} B = \frac{1}{4} B$

$\begin{aligned} ∴ B = B - \frac{1}{4} B \\ ∴ B = \frac{3}{4} B \end{aligned}$

Substituting the values,

$\begin{aligned} e & = \frac{Δ ϕ}{Δ t} \\ e & = \frac{A \times \frac{3}{4} B}{1} \\ ∴ e & = \frac{3}{4} AB \end{aligned}$

3. ⇒ (MHT CET 2023 10th May Evening Shift )

A hollow metal pipe is held vertically and bar magnet is dropped through it with its length along the axis of the pipe. The acceleration of the falling magnet is ( $g =$ acceleration due to gravity)

A. equal to g.

B. less than $g$ .

C. more than g.

D. zero.

Correct Option is (B)

As a bar magnet falls through the pipe, an emf is induced in the body of the pipe due to change in magnetic flux. However, the free fall of the magnet will be opposed due to the development of eddy currents in accordance with Lenz's law. This process continues till the bar magnet attains a constant terminal velocity.

4. ⇒ (MHT CET 2023 9th May Morning Shift )

The magnetic flux through a loop of resistance $10 Ω$ varying according to the relation $ϕ = 6 t^{2} + 7 t + 1$ , where $ϕ$ is in milliweber, time is in second at time $t = 1 s$ the induced e.m.f. is

A. $12 mV$

B. $7 mV$

C. $19 mV$

D. $19 V$

Correct Option is (C)

Given $R = 10 Ω, ϕ = 6 t^{2} + 7 t + 1 mWb, t = 1 s$

From $e = \frac{d ϕ}{dt}$ ,

$e = \frac{d}{d t} (6 t^{2} + 7 t + 1) = 12 t + 7$

Put $t = 1$ in the above equation,

$∴ e = 19 mV$

5. ⇒ (MHT CET 2021 20th September Morning Shift )

Two conducting wire loops are concentric and lie in the same plane. The current in the outer loop is clockwise and increasing with time. The induced current in the inner loop is

A. clockwise

B. anticlockwise

C. in a direction which depends on the ratio of the loop radii.

D. zero

Correct Option is (B)

Since the clockwise current is increasing, the induced current in the loop will be such as to oppose the change in magnetic flux and hence it will flow in anticlockwise direction.

⇒ Electromagnetic Induction