mathematical language

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Find the unit of the following quantities.  Express the units in terms of s (second for time), m (meter for length), N (Newton for force), C (Coulomb for electric charge).

  • Magnetic field B                                      
  • Magnetic auxiliary field H                      
  • Magnetic vector potential A                                
  • Magnetic dipole moment m                     
  • Magnetization M  
  • Magnetic inductance L
  • Product of B and H                      
  1. (20 points)

An equilateral triangle loop with side S lies in the xy plane, as shown in the figure below.  It carries a current I. 

  • Find the magnetic field produce by the loop at a point on the z axis which goes through the center of the triangle loop. The distance between the field point and the center of the loop is b, as shown in Fig. (a).

Hint: used the result for the magnetic field produced by a straight line current.

  • Calculate the magnetic dipole moment of the loop
  • When b>>S, using the dipole approximation, find the vector potential  and magnetic field  at the field point shown in Fig. (a). If the field point is moved off the z axis to a point as shown in Fig. (b).  The angle between the vector of the field point and the z axis is q, find the vector potential  and magnetic field
S
I
y
z
x
o
b
S
I
y
z
o
b
(a)
(b)
Field point
Field point
q
  • (20 points)

A metal bar of mass m slides frictionlessly down on two inclined parallel conducting rails a distance w apart, as shown in the figure below.  A resistor R is connected across the rails.  The resistance of the rails is 0.  A uniform magnetic field B is applied in the z direction (pointing up), which fills the entire region. The inclination angle is q.

  • There are two forces acting on the bar: the gravitational force and the magnetic force.  When the two forces balance each other, the bar is moving down at a constant speed v. Find the speed v, and emf  and the current I.
  • Find the heating power of the resistor
  • Find the decrease of gravitational potential energy per unit time.
x
w
R
y
z
v
q
I
  • (15 points)

A medium of a volume of 10m3 is in a magnetic field B of 1 Tesla.  The magnetic susceptibility  of the medium is 0.5. 

  • What is the magnetic auxiliary field H?
  • What the magnetization M?
  • What is the energy of the magnetic field W within the medium?
  • (15 points)

In a medium with an electric susceptibility , a magnetic susceptibility  and a conductivity , the electric field E and magnetic field B are functions of both position and time:  and .

  • What is the free electric current density ?
  • What is the polarization current density ?
  • What is the bound current density ?
  • What is the displacement current density ?

Among these four currents, which is produced by motion of charge carriers, and which is not?

6 (10 points)

A circular current loop lies in the xy plane.  The radius a of the circular loop is 1 micron.  The current I on the loop is 1 Ampere.  The loop is in a uniform magnetic field of 1 Tesla which makes the angle q of 45o with respect to the z axis as shown in the figure below. 

  • What is the force exerted on the loop by the magnetic field?
  • What is the torque exerted on the loop by the magnetic field?

No derivation of the formula is needed.

I
y
z
x
a
q