Question

The total charge induced in a conducting loop when it is moved in magnetic field depends on

• A. The rate of change of magnetic flux
• B. Initial magnetic flux only
• C. The total change in magnetic flux
• D. Final magnetic flux only

Answer 1

The correct option is C

The total change in magnetic flux

Step 1: Given data:

A conducting loop moving in a magnetic field

Let total charge $Q$

Step 2: Formula used:

Induced emf is given by-

$e=-\frac{d\varphi }{dt}\dots \dots \dots \dots \dots \dots \dots \left(1\right)$

Where, $\varphi$ is magnetic flux.

Ohm's law,

$V=IR\dots \dots \dots \dots \dots \dots \dots \left(2\right)$

Where, $V,I,R$ are voltage, current, and resistance respectively.

The total charge is given by-

$Q=\int Idt\dots \dots \dots \dots \dots \dots \dots \left(3\right)$ Where, $dt$ is the time interval in which current flow through the conducting loop

Step 3: Finding the relation between charge and flux:

From equation (2), if $e$is the emf then,

$e=IR$

Putting the value of $e$ from equation (1),

$\begin{array}{rcl}-\frac{d\varphi }{dt}& =& IR\\ I& =& -\frac{1}{R}\frac{d\varphi }{dt}\dots \dots \dots \dots \dots \dots \left(4\right)\end{array}$

The induced charge will be given by using the equation (3),

$Q=-{\int }_{{\varphi }_1}^{{\varphi }_2}\frac{1}{R}\frac{d\varphi }{dt}dt$

Thus, the induced charge,

$$\begin{gathered} Q=-\frac{1}{R}{\int }_{{\varphi }_1}^{{\varphi }_2}d\varphi \\ Q=\frac{1}{R}\left({\varphi }_1-{\phi }_2\right) \end{gathered}$$

Therefore, the total charge induced in a conducting loop when it is moved in magnetic field depends on the total change in magnetic flux.

Hence, option (C) is the correct answer.