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. d = R a) Using the Biot-Savart Law, determine the magnetic field along the z-axis. b) A uniform electric field is set up in the ŷ 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. 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?
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. d = R a) Using the Biot-Savart Law, determine the magnetic field along the z-axis. b) A uniform electric field is set up in the ŷ 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. 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?
College Physics
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Chapter1: Units, Trigonometry. And Vectors
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Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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Transcribed Image Text: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
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