dI/dt is the rate of change of current with respect to time, measured in amperes per second. In AP Physics C: E&M, it determines the voltage across an inductor through V_L = L(dI/dt), meaning an inductor pushes back against changing current, not current itself.
dI/dt is calculus notation for how fast the current in a circuit is changing at a given instant. Its units are amperes per second (A/s). On its own it's just a derivative, but in Unit 13 it becomes the star of the inductor equation V_L = L(dI/dt), where L is the inductance in henries.
Here's the idea that makes inductors click. A resistor cares about how much current flows (V = IR). An inductor doesn't care about the amount of current at all. It only cares about how fast the current is changing. A huge, steady current produces zero voltage across an ideal inductor because dI/dt = 0. A small current that's changing rapidly can produce a large voltage. Physically, this comes from Faraday's law. A changing current means a changing magnetic flux through the inductor's own coils, and that changing flux induces an emf that opposes the change (Lenz's law). That's why the equation is often written with a minus sign, ε = -L(dI/dt).
dI/dt lives in Topic 13.4 (Inductance) in AP Physics C: Electricity and Magnetism, and it's the bridge between the abstract idea of self-inductance and actual circuit behavior. This is also where the 'C' in Physics C earns its name. You'll be handed a current function like I(t) = I₀sin(ωt) and expected to differentiate it to get the inductor voltage, or handed a voltage and expected to reason backward about how the current must be changing. Every LR circuit, LC oscillation, and back-emf problem starts with V_L = L(dI/dt). If you understand what dI/dt is doing, the rest of Unit 13's circuit analysis follows.
Keep studying AP® Physics C: E&M Unit 13
Ideal Inductor (Unit 13)
The equation V_L = L(dI/dt) is the defining behavior of an ideal inductor. It tells you an ideal inductor acts like a plain wire when current is steady (dI/dt = 0 means V_L = 0) and resists sudden changes in current. That's why current through an inductor can never jump instantaneously. A jump would mean infinite dI/dt and infinite voltage.
Energy Stored in an Inductor (Unit 13)
While dI/dt sets the inductor's voltage, the current I itself sets its stored energy, U = ½LI². The two work together. The power flowing into the inductor is P = IV_L = LI(dI/dt), and integrating that power over time is exactly how you derive the ½LI² formula. It's a classic Physics C derivation to know cold.
Faraday's Law and Lenz's Law (Unit 13)
V_L = L(dI/dt) isn't a new law. It's Faraday's law applied to a coil's own magnetic field. The current creates flux proportional to I, so changing I changes flux, and ε = -dΦ/dt becomes ε = -L(dI/dt). Lenz's law supplies the minus sign, telling you the induced emf always fights the change in current.
Multiple-choice questions test whether you can actually take the derivative. A typical stem gives I(t) = I₀sin(ωt) and asks for the voltage across the inductor, expecting V_L = LI₀ω cos(ωt). A follow-up asks when the emf magnitude is maximum, and the answer is when the current crosses zero (cosine is maxed when sine is zero). That trips up anyone who assumes maximum current means maximum voltage. Numerical versions are common too. With L = 30 mH and I(t) = (2.0 A)sin(120πt), the maximum emf is LI₀ω = (0.030)(2.0)(120π) ≈ 23 V. Conceptual stems probe proportionality, like noticing that tripling dI/dt from 5 A/s to 15 A/s triples the voltage from 10 V to 30 V because inductance L is the constant of proportionality. On FRQs, dI/dt shows up inside LR circuit analysis, where you apply Kirchhoff's loop rule with the L(dI/dt) term and solve or interpret the resulting differential equation.
The current I tells you how much charge flows per second right now. dI/dt tells you how fast that flow is speeding up or slowing down. An inductor's voltage depends only on dI/dt, never on I alone. So an inductor carrying a large constant current has zero voltage across it, while an inductor at the instant its sinusoidal current passes through zero has maximum voltage. If you mix these up, you'll get sinusoidal AC questions exactly backward.
dI/dt is the instantaneous rate of change of current, measured in amperes per second.
The voltage across an inductor is V_L = L(dI/dt), so it depends on how fast the current changes, not on how big the current is.
For I(t) = I₀sin(ωt), the inductor voltage is LI₀ω cos(ωt), with maximum magnitude LI₀ω occurring when the current passes through zero.
Inductor voltage is directly proportional to dI/dt, and the inductance L is the constant of proportionality (tripling the rate triples the voltage).
Current through an inductor can never change instantaneously, because an instantaneous jump would require infinite dI/dt and infinite voltage.
The minus sign in ε = -L(dI/dt) comes from Lenz's law, since the induced emf opposes the change in current.
dI/dt is the rate of change of current with respect to time, in amperes per second. It matters because the voltage across an inductor equals L(dI/dt), so you find inductor voltage by differentiating the current function.
No, and this is the classic trap. For a sinusoidal current I(t) = I₀sin(ωt), the voltage LI₀ω cos(ωt) is maximum when the current is zero, because that's when the current is changing fastest. When current peaks, dI/dt = 0 and the inductor voltage is zero.
dq/dt is the current itself, the rate charge flows past a point. dI/dt is one derivative further, the rate that current is changing, which makes it the second derivative of charge. Resistor voltage uses I; inductor voltage uses dI/dt.
Differentiate to get dI/dt = I₀ω cos(ωt), so the inductor voltage is V_L = LI₀ω cos(ωt) with maximum magnitude LI₀ω. For example, with L = 30 mH and I(t) = (2.0 A)sin(120πt), the peak emf is (0.030)(2.0)(120π) ≈ 23 V.
Amperes per second (A/s). This is also where the henry comes from. Since V = L(dI/dt), one henry is one volt per ampere-per-second, so a 1 H inductor with current changing at 1 A/s has 1 V across it.
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