Question

Assertion $\left(A\right)$: The quantity $\left(\frac{L}{R}\right)$ possesses dimension of time.

Reason $\left(R\right)$: To reduce the rate of increase of current through a solenoid, we should increase the time constant $\left(\frac{L}{R}\right)$.

• A. Both $\left(A\right)$ and $\left(R\right)$ are true, and $\left(R\right)$ is the correct explanation of $\left(A\right)$
• B. Both $\left(A\right)$ and $\left(R\right)$ are true, but $\left(R\right)$ is not the correct explanation of $\left(A\right)$
• C. $\left(A\right)$ is true but $\left(R\right)$ is false
• D. $\left(A\right)$ is false but $\left(R\right)$ is true

Answer 1

The correct option is B Both $\left(A\right)$ and $\left(R\right)$ are true, but $\left(R\right)$ is not the correct explanation of $\left(A\right)$

Let us take a formula from $\text{LR}$ circuit as,

$I=I_0[1-e^{-\frac{t}{\left(\frac{L}{R}\right)}}]$

Since $\frac{t}{\left(\frac{L}{R}\right)}$ must be dimensionless

$\Rightarrow \frac{L}{R}$ must have dimension of time.

$\Rightarrow$ Assertion is true.

We know,

$I=\frac{E}{R}(1-e^{-\left(\frac{t}{\tau }\right)})[\because \tau =\left(\frac{L}{R}\right)=\text{Time constant}]$

$\frac{dI}{dt}=\frac{E}{R}\left(\frac{1}{\tau }{e}^{\left(\frac{-t}{\tau }\right)}\right)$

$\frac{dI}{dt}\propto \frac{1}{\tau }$

$\Rightarrow$ Reason is also correct but it does not explain the assertion.

<!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> Hence, $\left(B\right)$ is the correct answer.