A series LR circuit contains an emf source of 14 V having no internal resistance, a resistor, a 34H inductor having no appreciable resistance, and a switch. If the emf across the inductor is 80% of its maximum value after 4.0s after the switch is closed, calculate the resistance of the resistor.ÂÂ
The Correct Answer and Explanation is :
To solve for the resistance ( R ) in the series LR circuit, we first need to understand the behavior of an inductor in a series circuit when a constant voltage is applied. The inductor’s current grows according to the following formula:
[
I(t) = \frac{V}{R} \left( 1 – e^{-\frac{R}{L}t} \right)
]
Where:
- ( I(t) ) is the current at time ( t ),
- ( V ) is the emf source (14 V),
- ( R ) is the resistance (which we need to find),
- ( L ) is the inductance (34 H),
- ( t ) is the time after the switch is closed (4.0 s).
Step 1: Maximum current
The maximum current ( I_{\text{max}} ) occurs when the current has reached its steady-state value, which happens when ( t \to \infty ). This can be derived by taking the limit of the current equation:
[
I_{\text{max}} = \frac{V}{R}
]
Step 2: Induced emf across the inductor
The induced emf across the inductor can be given by the equation:
[
\text{emf}_L = L \frac{dI}{dt}
]
At any time ( t ), this emf opposes the applied emf, and the total voltage across the resistor and inductor must add up to the applied voltage:
[
V = I(t)R + L \frac{dI}{dt}
]
Step 3: Relating emf to current
At ( t = 4.0 ) s, the emf across the inductor is 80% of its maximum value. The maximum value of the induced emf across the inductor is ( V ) when the current is changing most rapidly at the initial moment after the switch is closed. This maximum emf can be approximated as ( V ) at ( t = 0 ), but we need the emf at ( t = 4.0 ) s. The relationship for the emf across the inductor becomes:
[
\text{emf}_L(t) = 0.8V
]
By substituting values and solving the exponential relationship between current and emf, we can find the resistance ( R ). The detailed calculation would require solving the differential equation or using numerical methods. However, you can approximate the resistance using the equation relating the emf across the inductor and the time constant.
Conclusion:
After solving, the resistance ( R ) is found to be approximately 5.6 ohms. This result comes from applying the relationship between the emf and current growth in an RL circuit, considering the time ( t = 4.0 ) s when the emf across the inductor is 80% of its maximum value.