A current `i_(0)` is flowing through an `L-R` circuit of time constant `t_(0)`. The source of the current is switched off at time `t = 0`. Let `r` be the value of `(-di//dt)` at time `t = 0`.Assuming this rate to be constant, the current will reduce to zero in a time interval of

Options

A. `t_(0)`
B. `et_(0)`
C. `(t_(0))/(e)`
D. `(1-(1)/(e))t_(0)`

Answer

To solve the problem, we need to analyze the behavior of the current in an L-R circuit when the source of current is switched off at time \( t = 0 \). The current \( i(t) \) in the circuit can be described by the following equation:

1. **Current Decay Equation**:
\[ i(t) = I_0 e^{-\frac{t}{\tau}} \]
where \( I_0 \) is the initial current, and \( \tau \) is the time constant of the circuit, given as \( t_0 \).

2. **Differentiating the Current**:
To find the rate of change of current, we differentiate \( i(t) \) with respect to time \( t \):
\[ \frac{di}{dt} = -\frac{I_0}{\tau} e^{-\frac{t}{\tau}} \]

3. **Evaluate at \( t = 0 \)**:
At \( t = 0 \), we substitute \( t = 0 \) into the differentiated equation:
\[ \frac{di}{dt}\bigg|_{t=0} = -\frac{I_0}{\tau} e^{0} = -\frac{I_0}{\tau} \]
This value is given as \( r \), thus:
\[ r = -\frac{I_0}{\tau} \]

4. **Finding the Time to Reduce Current to Zero**:
We want to find the time \( t_f \) when the current \( i(t_f) \) becomes zero. Setting the current equation to zero:
\[ 0 = I_0 e^{-\frac{t_f}{\tau}} \]
Since \( I_0 \) is not zero, we can ignore it and focus on the exponential term:
\[ e^{-\frac{t_f}{\tau}} = 0 \]
However, the exponential function never actually reaches zero. Instead, we can find the time it takes for the current to decay to a negligible amount.

If we assume the current reduces to zero in a time interval that is significant, we can express the time it takes for the current to decay to a very small value (for practical purposes, we can consider it to be when \( i(t) \) is approximately 0).

5. **Using the Relation Between \( r \) and \( \tau \)**:
From the earlier relation \( r = -\frac{I_0}{\tau} \), we can express \( \tau \) in terms of \( I_0 \) and \( r \):
\[ \tau = -\frac{I_0}{r} \]

6. **Final Time Calculation**:
The time taken for the current to reduce to zero can be approximated as:
\[ t_f = \tau = \frac{I_0}{r} \]

Thus, the time interval required for the current to reduce to zero is:
\[ \boxed{t_0} \]