The time constant of an R L circuit with L=25 mH is twice...

The time constant of an $R L$ circuit with $L=25 \mathrm{mH}$ is twice the time constant of an $R C$ circuit with $C=45 \mu F$. Both circuits have the same resistance $R$. Find (a) the value of $R$ and (b) the time constant of the $R L$ circuit.

Key Concepts

RL Circuit Time Constant

In an RL circuit, the time constant represents the time it takes for the current to reach approximately 63% of its final value after a step change in voltage. It is determined by the ratio of the inductance (L) to the resistance (R), given by the formula ?_RL = L/R.

RC Circuit Time Constant

In an RC circuit, the time constant indicates the time required for the capacitor to charge up to about 63% of the applied voltage or to discharge to about 37% of its initial voltage. It is calculated as the product of the resistance (R) and the capacitance (C), expressed as ?_RC = RC.

Relationship Between Time Constants

Comparing the time constants of different circuits is an important analytical technique in circuit analysis. In problems where a specified proportional relationship exists between the time constants of an RL and an RC circuit, the underlying formulas (L/R for RL and RC for RC) can be equated or related, leading to equations that allow solving for unknown circuit parameters.

Solving for Resistance in Circuit Analysis

When two circuits share the same resistor but have different time constant expressions, equating or relating their time constants enables one to isolate and compute the value of the resistance. This process often involves algebraic manipulation of the time constant formulas to express the resistance in terms of the known values of inductance and capacitance.

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