Question

In a coil of resistance $100\Omega$, a current is induced by changing the magnetic flux through it as shown in the figure. The magnitude of change in flux through the coil is:

• A. $200\mathrm{Wb}$
• B. $225\mathrm{Wb}$
• C. $250\mathrm{Wb}$
• D. $275\mathrm{Wb}$

Answer 1

The correct option is C

$250\mathrm{Wb}$

The explanation for correct answer:

Option (C) -

Step 1: Given data:

Resistance, $\mathrm{R}=100\Omega$

Current, $\mathrm{I}=10\mathrm{amp}$

Time, $\mathrm{T}=0.5\mathrm{s}$

Step 2: Formula used

EMF induced, $\mathrm{E}=\frac{\mathrm{d\varphi }}{\mathrm{dt}}$

Where, $\mathrm{\varphi }$ is the magnetic flux

EMF induced, $\mathrm{E}=\mathrm{IR}$

Where, $\mathrm{I}$ is current and $\mathrm{R}$ is resistance

Step 3: Finding the magnetic flux

By rearranging the formula for the induced emf, $\mathrm{E}=\frac{\mathrm{d\varphi }}{\mathrm{dt}}$

That is, $\mathrm{d\varphi }=\mathrm{Edt}....\left(1\right)$

EMF induced, $\mathrm{E}=\mathrm{IR}....\left(2\right)$

Substituting (2) in (1)

$$\begin{gathered} \mathrm{d\varphi }=\mathrm{IR}\mathrm{dt} \\ \mathrm{d\varphi }=\mathrm{R}\times \mathrm{Idt}....\left(3\right) \end{gathered}$$

Here, $\mathrm{Magnitude}\mathrm{of}\mathrm{change}\mathrm{in}\mathrm{flux}=\mathrm{Resistance}\times \mathrm{Area}\mathrm{under}\mathrm{current}\mathrm{v}/\mathrm{s}\mathrm{time}\mathrm{graph}$

The area under the current v/s time graph is = Area of the triangle formed.

$Idt=$Area under the current v/s time graph = Area of the triangle formed.

$$\begin{gathered} \Rightarrow Idt=\frac{1}{2}\times Base\times Height \\ \Rightarrow \frac{1}{2}\times t\times I \end{gathered}$$

$$\begin{gathered} \mathrm{d\varphi }=100\times \frac{1}{2}\times 0.5\times 10 \\ \mathrm{d\varphi }=250\mathrm{Wb} \end{gathered}$$

Therefore, the magnitude of change in flux through the coil is $250\mathrm{Wb}$.

Hence, option (C) is the correct answer.