Question

A Hall-effect probe to measure magnetic field strengths needs to be calibrated in a known magnetic field. Although it is not easy to do, magnetic fields can be precisely measured by measuring the cyclotron frequency of protons. A testing laboratory adjusts a magnetic field until the proton's cyclotron frequency is 9.60 MHz . At this field strength, the Hall voltage on the probe is 0.540 mV when the current through the probe is 0.143 mA . Later, when an unknown magnetic field is measured, the Hall voltage at the same current is 1.736 mV .A) What is the strength of this magnetic field?

Answer 1

Answer:

2.02395 T

Explanation:

q = Charge = $1.6\times 10^{-19}\ C$

m = Rest mass of proton = $1.67\times 10^{-27}\ kg$

f = Frequency = 9.6 MHz

I = Current

Cylclotron frequency is given by

$f=\frac{qB}{2\pi m}$

Magnetic field is given by

$B=\frac{2\pi mf}{q}\\\Rightarrow B=\frac{2\pi 1.67\times 10^{-27}\times 9.6\times 10^6}{1.6\times 10^{-19}}\\\Rightarrow B=0.62957\ T$

Hall voltage is given by

$v_h=\frac{IBD}{neA}$

For the two cases

$v_h_1=\frac{IB_1D}{neA}$

$v_h_2=\frac{IB_2D}{neA}$

Divinding the Hall voltages

$\frac{v_h_1}{v_h_2}=\frac{B_1}{B_2}\\\Rightarrow B_2=\frac{v_h_2}{v_h_1}\times B_1\\\Rightarrow B_2=\frac{1.736}{0.54}\times 0.62957\\\Rightarrow B_2=2.02395\ T$

The strength of the magnetic field is 2.02395 T