A resistor and inductor are connected in series through a battery.
The switch is closed at time t=0 . Then what is the magnitude of current flowing when rate of increases of magnetic energy in the inductor is maximum

Options

A. `I=V/R`
B. `I=V/(2R)`
C. `I=V/(4R)`
D. `I=V/R "in" 2`

Answer

To solve the problem step by step, we need to analyze the circuit consisting of a resistor (R) and an inductor (L) connected in series with a battery (V). We want to find the magnitude of the current flowing when the rate of increase of magnetic energy in the inductor is maximum.

### Step 1: Understand the Circuit
When the switch is closed at time \( t = 0 \), the current \( I(t) \) in the circuit increases over time according to the formula:
\[ I(t) = \frac{V}{R} \left(1 - e^{-\frac{R}{L}t}\right) \]
where \( V \) is the voltage of the battery, \( R \) is the resistance, and \( L \) is the inductance.

### Step 2: Magnetic Energy in the Inductor
The magnetic energy \( U \) stored in the inductor at any time \( t \) is given by:
\[ U = \frac{1}{2} L I^2 \]

### Step 3: Rate of Change of Magnetic Energy
To find the rate of change of magnetic energy, we differentiate \( U \) with respect to time \( t \):
\[ \frac{dU}{dt} = L I \frac{dI}{dt} \]

### Step 4: Find \( \frac{dI}{dt} \)
The derivative of the current \( I(t) \) with respect to time \( t \) is:
\[ \frac{dI}{dt} = \frac{V}{R} \cdot \frac{R}{L} e^{-\frac{R}{L}t} = \frac{V}{L} e^{-\frac{R}{L}t} \]

### Step 5: Substitute \( \frac{dI}{dt} \) into \( \frac{dU}{dt} \)
Substituting \( \frac{dI}{dt} \) into the expression for \( \frac{dU}{dt} \):
\[ \frac{dU}{dt} = L I \left(\frac{V}{L} e^{-\frac{R}{L}t}\right) = V I e^{-\frac{R}{L}t} \]

### Step 6: Find Maximum Rate of Change
To find when \( \frac{dU}{dt} \) is maximum, we need to express \( I \) in terms of \( e^{-\frac{R}{L}t} \):
\[ I = \frac{V}{R} (1 - e^{-\frac{R}{L}t}) \]
Substituting this into \( \frac{dU}{dt} \):
\[ \frac{dU}{dt} = V \left(\frac{V}{R} (1 - e^{-\frac{R}{L}t})\right) e^{-\frac{R}{L}t} \]
\[ = \frac{V^2}{R} (1 - e^{-\frac{R}{L}t}) e^{-\frac{R}{L}t} \]

### Step 7: Find the Condition for Maximum
To maximize \( \frac{dU}{dt} \), we set the derivative of \( \frac{dU}{dt} \) with respect to \( t \) to zero. After simplification, we find that the maximum occurs when:
\[ I = \frac{V}{2R} \]

### Conclusion
Thus, the magnitude of the current flowing when the rate of increase of magnetic energy in the inductor is maximum is:
\[ \boxed{\frac{V}{2R}} \]