Three problems of Electricity and Magnetism: Hall effect, Helmholtz coils and Ampere law. - 64953

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Problem 1

 

Consider Hall's experiment indicated in Fig. 1. The material is a metal with a density n of charge carriers each with charge q and mass m. The Hall voltage VH  is measured for a given applied field B current I and conductor thickness t. From these variables the so called Hall coefficient for the material is determined:

 

RH =VH t / IB      (1)

 

(a) For positive charge carriers, indicate on a sketch the carrier drift velocity vand which side of the conductor has positive and negative Hall voltage.

 

(b) For negative charge carriers, indicate on a sketch the carrier drift velocity and which side of the conductor has positive and negative Hall voltage.

 

(c) Show that the Hall coefficient RH = 1/nq. To do this you may need the following formulaethat you should derive or argue for: v = E/B, I = J w t, and      J = n q v .

 

(d) Calculate the Hall voltage for an experiment on a conductor where               n = 3.7×1022cm-3, q = 1.602×10-19C. The current I = 100 mA, the conductor thickness t = 0.1 mm, and the magnetic field B = 0.5 Tesla.

 

Figure 1: Experimental setup to measure the Hall coefficient for copper

 

 

 

 

 

Problem 2

 

Consider the Helmholtz coil configuration shown in Fig. 2. Here R = 0.2 m,

I = 10 A, and there are N = 100 windings in each coil. You can neglect the thickness of each coils windings so that its dimensions are fully specified by R.

 

(a) Use Biot and Savart's law to derive an expression for the magnetic field as a function of displacement  x  from the center of the coils.

 

(b) Calculate the strength of the magnetic field in Tesla at the central point between the coils where x = 0 and for x = ± R/2 and make a plot of B(x).

 

Figure 2: Helmholtz coil configuration.

 

 

 

 

 

 

Problem 3

 

Consider the coaxial conductor shown in Fig. 3. Assume the current I flows in opposite directions in the inner and outer conductor respectively. You can assume the current is uniformly distributed in the inner and outer conductors.

 

(a) Use Ampere's law to determine the magnetic field B within the inner conductor, between the conductors, and outside the outer conductor. Make a plot of B as a function of displacement from the center of the coax cable.

 

(b) Considering the Lorentz force on moving charge in a magnetic field, sketch the direction of forces acting on points along the circumference of the inner and outer conductor where current is flowing.

 

 

Figure 3: Coaxial cable with an inner and an outer.

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