Ch 30: Electromagnetic Induction

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 30: Electromagnetic Induction

Problem 69b

Chapter 30, Problem 69b

A 50 cm solenoid with 1000 turns has an inductance of 20 mH. Flipping a switch disconnects the inductor from the battery and connects it to a resistor. What is the value of the resistance if the magnetic field decreases by 50% in 150 μs?

Verified step by step guidance

Step 1: Understand the problem. The solenoid is disconnected from the battery and connected to a resistor, causing the magnetic field to decrease. The problem involves calculating the resistance using the time constant of the RL circuit. The time constant (τ) is related to the inductance (L) and resistance (R) by the formula:

τ = \frac{L}{R}

Step 2: Recall the relationship between the time constant and the decay of the magnetic field. The magnetic field decreases exponentially in an RL circuit, following the equation:

B = B

₀e^-tτ, where

B

₀ is the initial magnetic field,

B

is the magnetic field at time

t

, and

τ

is the time constant.

Step 3: Use the given information to find the time constant. The magnetic field decreases by 50% in 150 μs, meaning

B = \frac{1}{2} B

₀ at

t = 150 \times 10^- 6 s

. Substitute this into the exponential decay formula and solve for

τ

\frac{1}{2} = e^- \frac{150}{τ}

. Take the natural logarithm of both sides to isolate

τ

Step 4: Relate the time constant to the resistance. Once

τ

is calculated, use the formula

τ = \frac{L}{R}

to solve for

R

. The inductance

L

is given as 20 mH, or

20 \times 10^- 3 H

. Rearrange the formula to find

R = \frac{L}{τ}

Step 5: Substitute the values into the formula. Use the calculated value of

τ

and the given inductance

L

to find the resistance

R

. Ensure the units are consistent throughout the calculation.

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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 solenoid, that quantifies its ability to store energy in a magnetic field when an electric current flows through it. It is measured in henries (H) and is defined as the ratio of the induced electromotive force (emf) to the rate of change of current. In this question, the solenoid's inductance of 20 mH indicates how effectively it can store magnetic energy.

Time Constant

The time constant, denoted by τ (tau), is a measure of the time it takes for the current in an inductor to change significantly when connected to a resistor. It is calculated as τ = L/R, where L is the inductance and R is the resistance. In this scenario, the time constant will help determine how quickly the magnetic field decreases when the inductor is disconnected from the battery and connected to the resistor.

Magnetic Field Decay

When an inductor is disconnected from a power source and connected to a resistor, the magnetic field associated with the inductor begins to decay. The rate of this decay is exponential, characterized by the time constant. In this question, the magnetic field decreases by 50% in 150 μs, which can be used to find the resistance by relating the decay time to the time constant of the circuit.

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Textbook Question

BIO One possible concern with MRI (see Exercise 28) is turning the magnetic field on or off too quickly. Bodily fluids are conductors, and a changing magnetic field could cause electric currents to flow through the patient. Suppose a typical patient has a maximum cross-section area of 0.060 m². What is the smallest time interval in which a 5.0 T magnetic field can be turned on or off if the induced emf around the patient's body must be kept to less than 0.10 V?

Textbook Question

A 50 cm solenoid with 1000 turns has an inductance of 20 mH. What is the magnetic field strength inside the inductor when the current is 75 mA?

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Textbook Question

CALC FIGURE P30.67 shows the potential difference across a 20 mH inductor. The current through the inductor at t = 0 ms is 0.25 A. What is the current at t = 10 ms?

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Textbook Question

CALC The current through inductance L is given by $I = I_{0} e^{- t / τ}$ . Find an expression for the potential difference ΔV_L across the inductor.

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Textbook Question

An LC circuit is built with a 20 mH inductor and an 8.0 pF capacitor. The capacitor voltage has its maximum value of 25 V at t = 0 s. What is the inductor current at that time?

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Textbook Question

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.