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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 18.00 cm, each coil carries a current of 7.00 A and have 420.0 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, mu0. Evaluate the magnetic field numerically for the given locations.

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Ans:- Biot-Savart Law:
dB = (?0/4..3.14).(I/r^2).(dl cross product r)

Now apply the Biot-Savart Law:
Now we integrate over dl, but x, R, and cos (phi) are all con
This can be resolved into two components, one along the axis OP, and other PS, which is perpendicular to OP. PS is exactly cancelled by the perpendicular component PS(?) of the field due to a current and centered at A(?).So, the total magnetic field at a point which is at a distance x away from the axis of a circular coil of radius r is given by,
If there are n turns in the coil, then
this equation is for one loop Bx = 103.9/(x^2 + r^2)^3/2 * 10^-6 tesla
=> the equation remain same for second loop but for right side of both loop
x will be x-R
net magnetic field at right side of loops = 103.9*10^-6 /(x^2 + r^2)^3/2 + 103.9*10^-6 / (x^2 + 2.r^2 -2.x.r) tesla
similarly for left and in between these loops magnetic field can be obtained .
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