CALC The current through inductance L is given by I=I0e−t/τI = $$I_0$$ $$e^{-t/\tau}. $$Evaluate ΔVL at t = 0, 1.0, and 3.0 ms if L = 20 mH, I0 = 50 mA, and τ\tau = 1.0 ms.

Ch 30: Electromagnetic Induction

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 30: Electromagnetic Induction

Problem 70b

Chapter 30, Problem 70b

CALC The current through inductance L is given by $I = I_{0} e^{- t / τ}$ . Evaluate ΔV_L at t = 0, 1.0, and 3.0 ms if L = 20 mH, I₀ = 50 mA, and $τ$ = 1.0 ms.

Verified step by step guidance

Step 1: Recall the formula for the voltage across an inductor, which is given by ΔV_L = L * (dI/dt), where L is the inductance and dI/dt is the time derivative of the current I.

Step 2: Write the given current equation I(t) = I_0 * e^(-t/τ). Differentiate this equation with respect to time t to find dI/dt. The derivative is dI/dt = -I_0/τ * e^(-t/τ).

Step 3: Substitute the given values into the expression for dI/dt. Use I_0 = 50 mA (0.050 A), τ = 1.0 ms (0.001 s), and L = 20 mH (0.020 H). The expression becomes ΔV_L = L * (-I_0/τ * e^(-t/τ)).

Step 4: Evaluate ΔV_L at the specified times t = 0, 1.0 ms, and 3.0 ms. For each time, substitute t into the expression ΔV_L = 0.020 * (-0.050/0.001) * e^(-t/0.001). Simplify the exponential term e^(-t/0.001) for each time value.

Step 5: After substituting and simplifying for each time value, you will have the voltage across the inductor ΔV_L at t = 0, 1.0 ms, and 3.0 ms. Ensure the units are consistent throughout the calculation (e.g., volts for ΔV_L).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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). In circuits, inductance affects how quickly current can change, influencing the behavior of the circuit during transient states.

Time Constant (τ)

The time constant, denoted as τ (tau), is a measure of the time it takes for the current in an inductor to reach approximately 63.2% of its maximum value after a change in voltage. It is calculated as τ = L/R, where L is inductance and R is resistance. In this context, τ helps determine how quickly the current decays over time.

Voltage Across an Inductor (ΔVL)

The voltage across an inductor (ΔVL) is related to the rate of change of current through it. According to Faraday's law of electromagnetic induction, this voltage can be expressed as ΔVL = L * (dI/dt), where dI/dt is the rate of change of current with respect to time. Understanding this relationship is crucial for evaluating the voltage at specific time intervals in the given problem.

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CALC The current through inductance L is given by I=I0e−t/τI = I_0 e^{-t/\(\tau\)}. Evaluate ΔVL at t = 0, 1.0, and 3.0 ms if L = 20 mH, I0 = 50 mA, and τ\(\tau\) = 1.0 ms.

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Key Concepts

Inductance

Time Constant (τ)

Voltage Across an Inductor (ΔVL)

CALC The current through inductance L is given by $I = I_{0} e^{- t / τ}$ . Evaluate ΔV_L at t = 0, 1.0, and 3.0 ms if L = 20 mH, I₀ = 50 mA, and $τ$ = 1.0 ms.