Question

One practical way to measure magnetic field strength uses a small, closely wound coil called a search coil. The coil is initially held with its plane perpendicular to a magnetic field. The coil is then either quickly rotated a quarter-turn about a diameter or quickly pulled out of the field. (a) Derive the equation relating the total charge Q that flows through a search coil to the magnetic-field magnitude B. The search coil has N turns, each with area A, and the flux through the coil is decreased from its initial maximum value to zero in a time ∆t. The resistance of the coil is R, and the total charge is Q = I∆t, where I is the average current induced by the change in flux. (b) In a credit card reader, the magnetic strip on the back of a credit card is rapidly “swiped” past a coil within the reader. Explain, using the same ideas that underlie the operation of a search coil, how the reader can decode the information stored in the pattern of magnetization on the strip. (c) Is it necessary that the credit card be “swiped” through the reader at exactly the right speed? Why or why not?

Short answer

1. The total charge flow is$Q=\frac{NBA}{R}$
2. The magnetic stripe on a credit card is made up of a pattern of magnetic fields, so when the card is passed through the reader, it causes a pattern of charges to flow in the reader coil, allowing the reader to read the card's information.
3. The charge depends only on the change in the magnetic flux, according to the result of part(a), and not on the time of the flux change.

Step-by-step solution

Magnetic flux

Magnetic flux is expressed as;

$\mathrm{\Phi }=\mathrm{Bacos}\left(\mathrm{\varphi }\right)$

So, here B is the magnetic field

A is area.

Total charge flow

b. Consider a search coil, which is used to determine the strength of a magnetic field. The coil's plane is initially held perpendicular to a magnetic field. The coil is then quickly rotated a quarter turn around a diameter or quickly pulled out of the field, reducing the flux through the coil from its maximum value to zero in a short period of time.

$$\begin{gathered} {\Phi }_{B,i}=BA\cos \left(\varphi \right)=BA \\ {\Phi }_{B,f}=0 \end{gathered}$$

The magnitude of the average emf induced in the coil equals the flux change divided by the time required to go from the initial flux to the final flux and multiplied by the number of turns N;

$\left|\mathrm{\epsilon av}\right|=\mathrm{N}\frac{\left|{\mathrm{\Phi }}_{\mathrm{Bf}}-{\mathrm{\Phi }}_{\mathrm{Bi}}\right|}{∆\mathrm{t}}=\frac{\mathrm{NBA}}{∆\mathrm{t}}$

the induced current equals the induced emf divided by the resistance of the search coil, that is,

$l=\frac{\left|\epsilon =av\right|}{R}=\frac{NBA}{R∆t}$

So, the total charge flow is;

localid="1664167799568" $$\begin{gathered} Q=l∆t=\left(\frac{NBA}{R∆t}\right)∆t=\frac{NBA}{R} \\ Q=\frac{NBA}{R} \end{gathered}$$

Hence, the total charge flow is$\frac{NBA}{R}$

Operation of a search coil

b. The magnetic stripe on a credit card is made up of a pattern of magnetic fields, so when the card is passed through the reader, it causes a pattern of charges to flow in the reader coil, allowing the reader to read the card's information.

Swipe Speed

c. Hence, the charge depends only on the change in the magnetic flux, according to the result of part(a), and not on the time of the flux change.