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Chapter 27: Problem 51

A flip coil is used to measure magnetic fields. It's a small coil placed with its plane perpendicular to a magnetic field, and then flipped through \(180^{\circ} .\) The coil is connected to an instrument that measures the total charge \(Q\) that flows during this process. If the coil has \(N\) turns, area \(A,\) and resistance \(R,\) show that the field strength is \(B=Q R / 2 N A.\)

Short Answer

Expert verified
The magnetic field strength \( B \) is given by \( B= QR/2NA \).

Step by step solution

01

Understand and apply Faraday's Law

Start by using Faraday's law. According to Faraday's Law, the induced emf \( E \) in a closed circuit is equal to the rate of change of magnetic flux \( \Phi \) through the circuit. In mathematical term: \( E = -d\Phi/dt \)
02

Calculate total charge \( Q \)

The total charge that the instrument measures is the time integral of the current, since by definition, current \( I \) is the amount of the charge \( Q \) flowing per time \( t \). This gives: \( Q = \int Idt = \int Edt/R \), where \( R \) is the coil resistance.
03

Compute magnetic flux change \( \Delta \Phi \)

The change in magnetic flux \( \Delta \Phi \) when the coil is flipped 180 degrees is \( \Delta \Phi = 2BA \), where \( B \) is the field strength and \( A \) is the area of the coil.
04

Combine the equations

Substitute the change in magnetic flux \( \Delta \Phi \) into the equation from Step 1 giving \( E = -\Delta \Phi / \Delta t \). The EMF is also equal to \( Q / R\Delta t \) from Step 2. Setting these equal gives \( Q / R\Delta t = -\Delta \Phi / \Delta t \). Solve for \( B \) to give the final formula.

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Most popular questions from this chapter

A single-turn loop of radius \(R\) carries current \(I\). How does the magnetic- energy density at the loop center compare with that of a long solenoid of the same radius, carrying the same current, and consisting of \(n\) turns per unit length?

Can an induced electric field exist in the absence of a conductor?

Find the magnetic-field strength in a region where the magneticenergy density is \(7.8 \mathrm{J} / \mathrm{cm}^{3} .\)

A wire of radius \(R\) carries current \(I\) distributed uniformly over its cross section. Find an expression for the total magnetic energy per unit length within the wire.

A generator consists of a rectangular coil \(75 \mathrm{cm}\) by \(1.3 \mathrm{m},\) spinning in a 0.14 -T magnetic field. If it's to produce a \(60-\mathrm{Hz}\) alternating emf with peak value \(6.7 \mathrm{kV},\) how many turns must it have?
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