CALC The current through inductance L is given by $I=I_0{e}^{-t/\tau }$. Find an expression for the potential difference ΔV_L across the inductor.

Verified step by step guidance

1. The potential difference across an inductor is given by the formula: $∆V_L=\; -L\frac{\mathrm{dI}}{\mathrm{dt}}$, where $L$ is the inductance and $I$ is the current through the inductor.
2. Substitute the given expression for current $I=I_0{e}^{-\frac{t}{\tau }}$ into the formula for $I$.
3. Differentiate the current $I=I_0{e}^{-\frac{t}{\tau }}$ with respect to time $t$. The derivative is: $\frac{\mathrm{dI}}{\mathrm{dt}}=\; -I_0\frac{1}{\tau }{e}^{-\frac{t}{\tau }}$.
4. Substitute the derivative $\frac{\mathrm{dI}}{\mathrm{dt}}$ into the formula for the potential difference: $∆V_L=\; -L\frac{\mathrm{dI}}{\mathrm{dt}}$.
5. Simplify the expression to get the final result: $∆V_L=\; L{I}_0\frac{1}{\tau }{e}^{-\frac{t}{\tau }}$.

Key Concepts

Inductance

Inductance is a property of an electrical component, typically a coil or inductor, that quantifies its ability to store energy in a magnetic field when an electric current flows through it. The unit of inductance is the henry (H). Inductance is crucial in understanding how inductors respond to changes in current, as it determines the induced electromotive force (emf) that opposes changes in current according to Lenz's law.

Ohm's Law

Ohm's Law states that the current (I) flowing through a conductor between two points is directly proportional to the voltage (V) across the two points and inversely proportional to the resistance (R) of the conductor. This relationship is expressed as V = IR. Understanding Ohm's Law is essential for analyzing circuits, especially when calculating the potential difference across components like inductors.

Electromotive Force (emf)

Electromotive force (emf) is the energy provided per charge by a source of electrical energy, such as a battery or an inductor. In the context of inductors, the emf is induced due to a change in current, as described by Faraday's law of electromagnetic induction. The potential difference across an inductor can be expressed as ΔV_L = -L(dI/dt), where L is the inductance and dI/dt is the rate of change of current, highlighting the relationship between current change and induced voltage.