At the instant t = 0 when a battery of voltage V is connected to a series RL circuit (initially at rest), what is the voltage across the inductor?
AZero — current hasn't flowed yet so the inductor is inactive
BV/2 — the voltage is split equally between the inductor and resistor at t = 0
CV — all of the battery voltage appears across the inductor because current is zero and V_R = IR = 0
DV/R — the same as the steady-state current times resistance
At t = 0, the inductor enforces I = 0 (it cannot allow an instantaneous current jump). Since V_R = IR = 0·R = 0, Kirchhoff's voltage law (V = V_R + V_L) gives V_L = V − 0 = V. All of the battery voltage is initially across the inductor. As current builds and I increases, V_R grows and V_L shrinks, until at steady state V_L = 0 and V_R = V. The transient is a smooth handoff of voltage from inductor to resistor.
Question 2 Multiple Choice
An engineer doubles the inductance L in an RL circuit while keeping resistance R constant. What happens to the time constant τ and the speed at which current approaches its final value?
Aτ is halved — larger inductance means the circuit reaches steady state faster
Bτ is unchanged — only R affects the time constant
Cτ is doubled — the inductor fights harder against current changes, so the rise is slower
Dτ is doubled and the final current I = V/R is also doubled
τ = L/R, so doubling L doubles τ. A larger inductance means greater opposition to current changes (larger back-EMF for the same dI/dt), so it takes longer for current to build to its final value. The final current V/R is unchanged — it depends only on resistance. After one τ, current reaches 63% of V/R; after 5τ, it is at 99%. Doubling τ means it takes twice as long to reach each milestone. Larger L = slower transient; larger R = faster transient (τ = L/R decreases).
Question 3 True / False
A larger resistance R in a series RL circuit causes the current to rise more slowly to its final steady-state value.
TTrue
FFalse
Answer: False
Larger R actually speeds up the transient. The time constant is τ = L/R, so increasing R decreases τ, meaning the current approaches its final value faster. Intuitively, higher R dissipates energy more rapidly, reducing the inductor's ability to sustain its opposition. The final current V/R is also smaller (since R is larger), but it is reached in fewer seconds. Many students expect larger R to slow things down because it 'resists' current — but the time constant τ = L/R shows R appears in the denominator.
Question 4 True / False
At steady state in a DC RL circuit, the inductor carries the full steady-state current and behaves effectively as a short circuit (a plain wire).
TTrue
FFalse
Answer: True
At steady state, dI/dt = 0 because current is no longer changing. The inductor's voltage is V_L = L dI/dt = L·0 = 0. An element with zero voltage drop and nonzero current through it is electrically equivalent to a wire (short circuit). All of the battery voltage then appears across the resistor, and I = V/R. This is why inductors 'look like wires' to DC at steady state and why they are used to block AC signals while passing DC — at steady DC, they disappear electrically.
Question 5 Short Answer
Why is the initial current in a DC RL circuit exactly zero when voltage is first applied, and why does the final current equal V/R rather than some other value?
Think about your answer, then reveal below.
Model answer: The initial current is zero because the inductor enforces continuity of current — any instantaneous jump would require infinite voltage (V_L = L dI/dt with dI/dt → ∞). The inductor opposes the change, so current starts at zero and builds gradually. The final current equals V/R because at steady state the current is no longer changing (dI/dt = 0), so the inductor's voltage drop V_L = L dI/dt = 0, and Kirchhoff's law gives V = IR, so I = V/R.
These two boundary conditions — I(0) = 0 and I(∞) = V/R — fully determine the exponential solution I(t) = (V/R)(1 − e^(−t/τ)). The initial condition comes from the inductor's physical constraint (current continuity); the final condition comes from the DC steady state where inductance is irrelevant. The exponential curve with time constant τ = L/R smoothly interpolates between these two values. Understanding the physics at t = 0 and t → ∞ is the key to understanding the transient at all intermediate times.