Question

A long solenoid of diameter 0.1m has 2×10⁴ turns per meter. At the center of the solenoid, a coil of 100 turns and a radius of 0.01 m is placed with its axis coinciding with the solenoid axis. The current in the solenoid reduces at a constant rate to 0A from 4A in 0.05s. If the resistance of the coil is 10π²Ω the total charge flowing through the coil during this time is?

• A. 32 πμC
• B. 32 μC
• C. 16 πC
• D. 16 πμ

Answer 1

The correct option is B

32 μC

Step 1: Given data

Number of turns, $N=100$

Radius, $r=0.01m$

Resistance, $R=10{\mathrm{\pi }}^2\mathrm{\Omega }$

Step 2: Formula used

According to Lenz's law the induced EMF is $\epsilon =-N\frac{d\varphi }{dt}$,

(where $N$ is number of turns in the coil, $\varphi$ is magnetic flux and $t$ is time)

Current is given by $I=\frac{\epsilon }{R}$

(where $\epsilon$ is induced EMF and $R$ is resistance)

Current $\left(I\right)$ is the rate of flow of charge $\left(q\right)$ in time $\left(t\right)$: $I=\frac{dq}{dt}$.

Magnetic flux is given by $\varphi =BA$, where $B$ is magnetic field and $A$ is area of cross-section of solenoid.

Magnetic field is given by $B={\mu }_{0}l$, where ${\mu }_0=4\mathrm{\pi }\times {10}^{-7}$ is a constant and $I$ is the current.

Step 3: Solution

Now, on dividing by $R$ on both sides,

$$\begin{gathered} \frac{\epsilon }{R}=-\frac{N}{R}\frac{d\varphi }{dt} \\ ∆I=-\frac{N}{R}\frac{d\varphi }{dt} \\ \frac{∆q}{∆t}=-\frac{N}{R}\frac{∆\varphi }{∆t} \\ ∆q=-\left[\frac{N}{R}\left(\frac{∆\varphi }{∆t}\right)\right]∆t \end{gathered}$$

Here, the $-ve$ sign shows that induced EMF opposes the change in flux.

$$\begin{gathered} ∆q=-\frac{N∆\varphi }{R} \\ =-\frac{N∆BA}{R} \\ =-\frac{NA∆B}{R} \\ =-\frac{NA{\mu }_0∆l}{R} \end{gathered}$$

Substitute the values in the above expression obtained

$$\begin{gathered} ∆q=\frac{4\mathrm{\pi }\times {10}^{-7}\times 100\times 4\times \mathrm{\pi }\times {(0.01)}^2}{10{\mathrm{\pi }}^2} \\ ∆q=32\mu C \end{gathered}$$

Therefore, the total charge flowing through the coil during this time is, $∆q=32\mu C$

Hence, option (B) is the correct option.