Question

Helmholtz coils are frequently used in experiments because they create a uniform magnetic field about their axial direction. For example, they are traditionally used in the experiment for the measurement of the charge to mass ratio of the electron. Helmholtz coils are two coils separated by a distance equal to their radius and also carry equal currents in the same direction, as shown in the figure on the right. Suppose the radius of the coils is 12.00 cm, each coil carries a current of 11.50 A and have 460.0 turns. The coils are in the yz plane where one coil\'s center is at x = 0 the other coil\'s center is at x = R. Using the Biot-Savart law, what is the expression for the resultant magnetic field, Bx, along the line x joining their centers. Express the magnetic field in terms of the radius of the coils R, the number of turns N, the current I, x, and the permeability of free space, μ0.

Also,

B_x(3.000) = _____

B_x(9.000) = _____

Answer 1

given:

radius of the coil=R=0.12 m

current=I=11.5 A

number of turns=N=460

field on the axis of a current carrying coil=mu*I*N*R^2/(2*(x^2+R^2)^1.5)

where z is the distance of the point from the center of the coil.

for 0<x<R:

field direction due to the coil at x=0 will be in opposite direction of field due to coil at x=R

total field=(mu*I*N*R^2/(2*(x^2+R^2)^1.5))-(mu*I*N*R^2/(2*((R-x)^2+R^2)^1.5))

for x>R:

field due to both the coils are in same direction.

total field=(mu*I*N*R^2/(2*(x^2+R^2)^1.5))+(mu*I*N*R^2/(2*((x-R)^2+R^2)^1.5))

part 2:

as x=3 is in between x=0 and x=R:

total field=(mu*I*N*R^2/(2*(x^2+R^2)^1.5))-(mu*I*N*R^2/(2*((R-x)^2+R^2)^1.5))

using the given values,

total field=0.0111 Wb/m^2

part 3:

at x=9 cm=0.09 m,

total field=(mu*I*N*R^2/(2*(x^2+R^2)^1.5))-(mu*I*N*R^2/(2*((R-x)^2+R^2)^1.5))

=-0.0111 Wb/m^2