A \"write head\" on a computer hard drive used currents in wire loops to produce

Question

A "write head" on a computer hard drive used currents in wire loops to produce strong magnetic fields to alter the magnetic structure of the disc and thereby encode information. The "read head" senses changes in the magnetic field of the disc and converts them to electrical currents that carry the encoded information. A model of a write head is a length of conducting wire wrapped into a tightly wound spiral. This can be approximated by a series of N concentric circles, each separated by the same distance. Derive an expression for the magnetic field at the center of a loop of uniform radius [r] that carries a current [I]. Derive an expression for the magnetic field at the center of N concentric circles separated by the same distance, that carry the same current [I] in the direction.

Explanation / Answer

For this problem we can calculate the magnetic field at the center of a loop and then sums the field each turn

Part a)

We calculate the magnetic field of a current loop at its center

We use the law Biot- Savat

dB = o I/4 ds x r / R2 r unit vector

by circular symmetry the resulting field must be perpendicular to the ring

dB = o I/4 ds/ R2

If we locate a coordinate system with the x-axis perpendicular the ring, the resulting component isBx = o I/4 ds Cos /(x2 + R2)

cos = R/ (x2 +R2)1/2

Bx = oI/4 R/(x2+R2)3/22R

in the center of the ring x= 0

Bx = o I/2 R2 /(R2)3/2 =o I/2 1/R

The camp in the center of the ring is

Bx = o I/2R

Part b)

Each of the rings creates a field with the same sense and the only change is the radius

Bx1 = o I/2R1

Bx2 = o I/2R2

Bx3= o I/2R3

Bt = Bx1+Bx2+Bx3+ ….+ Bxn

Bt = o I/2 ( 1/R1 +1/R2+ 1/R3+ … +1/Rn)

as each ring is separated the same distance

R1 = 1R

R2 = 2R

R3 = 3R

R4 =4 R

Rn = n R

( 1/R1 +1/R2+ 1/R3+ … +1/Rn) = 1/R (1+1/2+1/3+1/4+… 1/n) = 1/k

This is a harmonic series

Bt = oI/2R ( 1 +1/2+ 1/3+ … +1/n)

Bt = o I/2R log(n)

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