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 51

Chapter 30, Problem 51

A small, 2.0-mm-diameter circular loop with R = 0.020 Ω is at the center of a large 100-mm-diameter circular loop. Both loops lie in the same plane. The current in the outer loop changes from +1.0 A to −1.0 A in 0.10 s. What is the induced current in the inner loop?

Verified step by step guidance

Determine the magnetic field at the center of the large loop due to the current in the outer loop. Use the formula for the magnetic field at the center of a circular loop:

B = μ

₀I2R, where

μ

₀ is the permeability of free space,

I

is the current, and

R

is the radius of the loop.

Calculate the change in magnetic flux through the smaller loop. The magnetic flux is given by

Φ = B A

, where

B

is the magnetic field and

A

is the area of the smaller loop. The area can be calculated using

A = π r

², where

r

is the radius of the smaller loop.

Determine the rate of change of magnetic flux through the smaller loop. The rate of change of flux is given by

ΔΦ Δt

, where

ΔΦ

is the change in flux and

Δt

is the time interval over which the change occurs.

Use Faraday's law of electromagnetic induction to calculate the induced emf in the smaller loop. Faraday's law states that the induced emf is equal to the negative rate of change of magnetic flux:

ε = - ΔΦ Δt

. Ignore the negative sign for magnitude calculations.

Calculate the induced current in the smaller loop using Ohm's law:

I = ε R

, where

ε

is the induced emf and

R

is the resistance of the smaller loop.

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Key Concepts

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

Faraday's Law of Electromagnetic Induction

Faraday's Law states that a change in magnetic flux through a loop induces an electromotive force (EMF) in that loop. The induced EMF is proportional to the rate of change of the magnetic flux. In this scenario, the changing current in the outer loop alters the magnetic field, which affects the inner loop, leading to an induced current.

Magnetic Flux

Magnetic flux is defined as the product of the magnetic field strength and the area through which the field lines pass, taking into account the angle between the field lines and the normal to the surface. It quantifies the total magnetic field passing through a given area. In this problem, the magnetic flux through the inner loop changes as the current in the outer loop varies.

Ohm's Law

Ohm's Law relates the voltage (V), current (I), and resistance (R) in an electrical circuit, expressed as V = IR. When an induced EMF is generated in the inner loop due to the changing magnetic flux, Ohm's Law can be used to calculate the induced current by dividing the induced EMF by the resistance of the inner loop.

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