E&M-Advanced Physics Question

Question

A Helmholtz coil is a pair of circular wire loops that produce a uniform magnetic field in the center. Consider two circular loops of wire with radius R, with each loop centered on the z - axis (at z = ±d/2), where the separation d is the same as the radius R of the loops. A current I is circulating in the + direction in both wire loops.

a) Using the Biot-Savart Law, determine the magnetic field along the z-axis.

b) A uniform electric field is set up in the direction to allow a beam of electrons moving in the direction to pass through the uniform magnetic field undeflected. If the magnitude of the electric field is kept the same, but the direction was changed to the -ŷ direction, describe the trajectory of the electron beam as it passes through the Helmholtz coil. If possible, provide a mathematical equation describing the general trajectory. d=R 16 z=0

c) The direction of the current in the bottom of loop is reversed, as shown in the second figure. Far from the pair of loops, how does the strength of the magnetic field produced by this system depend on the distance away? Hint: What is the first non-zero term in the monopole expansion for the vector potential? d=R 12=0

Answer 1

Bio-Savat law provides that magnetic field B on the axis of the coil of radius R (x-axis) is

B = μ₀ I R² / 2 (x² + R² )^3/2

where x = 0 at the center of the coil.

Consider top coil, offset from the origin by d/2 = R/2 along z axis. Coil axis is z. Therefore variable x in equation above should be replaced with (R/2 - z). To verify: when z = +R/2, i.e. at the center of the top coil, x-term in the 1st equation disappears. Good. Therefore at some point on z-axis magnetic field (B_z component) generated by the top coil is:

B_top(z) = μ₀ I R² / 2 [ ( R/2 - z )² + R² ]^3/2

^{Similarly, but slightly different, the bottom coil generates filed B}_bot ^{at the same point on z-axis}

B_bot(z) = μ₀ I R² / 2 [ ( R/2 + z )² + R² ]^3/2

^{Total field B is a superposition of B}top ^{and B}bot^.

^{Although B is a vector, on z-axis Bx = By = 0 i.e.}

^Btotal ^{= B}top ^{+ B}bot

Which at z = 0 is more-or-less uniform, i.e. not a function of z. And this was the intention of inventor, Dr. Helmholtz.

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