Helmholtz coils are frequently used in experiments because they create a uniform

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 19.00 cm, each coil carries a current of 5.00 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, 0.

Evaluate the magnetic field found above at x = 4.750 cm (1/4R) and x = 14.25 cm (3/4R).

Explanation / Answer

We know that the resultant magnetic field of the helmhotz coil is given by
B = 2B1(x)
Where B1(x) is the magnetic field due to one coil
B1(x) = (uonIR2 )/2(R2 + x2)3/2
Hence the net magnetic field will be
B = (uonIR2 )/(R2 + x2)3/2
and the direction will be toward +X direction .
Now the magentic field at X = 4.75 cm
B = (4Pi*10-7*460*5*(0.19)2) /(0.192 + 0.04752)3/2
B = (208.677*10-7) /(0.0075) = 27.779*10-4 T
Magnetic field at X = 14.25 cm
B = (4Pi*10-7*460*5*(0.19)2) /(0.192 + 0.14752)3/2
B = (208.677*10-7) /(0.0139) = 14.995*10-4 T

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