Exam 2 Solutions

Exam 2 Solutions

1. [8 points] In the circuit shown, the resistance. The battery voltages are identical:. What is the current (in amps) flowing through the middle branch from a to b?

Applying the junction rule:

Applying the loop rule to the left loop (LL), right loop (RL), and the full loop (FL) gives:

Plugging the FL equation into the junction rule equation, and then into the RL equation gives:

Thus no current flows from top to bottom in the middle branch (or anywhere for that matter). Since all batteries have a potential difference of 1 V, one can see that the loop rule works out if all currents are zero.

2. [6 points] A capacitor of capacitance contains a charge q on one of its plates (and –q on the other). It is connected in to a resistor of resistance R = 200W such that it forms a closed circuit. How much time must elapse so that the charge on the capacitor is reduced to only 10% of its starting value?

Applying Kirchoff’s loop rule, the equation to solve is:

Thus the elapsed time so that 10% of the charge remains is determined by:

3. [6 points] A stationary flat conductor carries a constant current i >0 in the direction from top to bottom () in the presence of a magnetic field B that points into plane the paper (). If the electrons that make up the current are not allowed to leave the conductor, and the magnitude of their drift velocity is , indicate the direction and determine the magnitude of any electric field created inside the conductor.

4. [6 points] An electron is accelerated from rest between 2 electric plates with a potential difference of . What is the velocity of the electron upon reaching the far plate? The mass of the electron is .

The potential energy of the electron when in starts from rest is translated into kinetic energy as it approaches the second plate:

We need to use special relativity here because the energy will be comparable to the rest mass energy.

5. [6 points] A long wire has an electric charge density of , where , as measured in the rest frame of the wire. What is the magnitude of the electric field in the rest frame of an electron traveling at a velocity of parallel to the wire at a distance from the wire?

In the rest frame of the wire, and by using Gauss’ Law, we can solve for the electric field:

In the rest frame of the electron, however, the wire appears length contracted (since the electron’s motion is parallel to the wire). Thus the charge per unit observed length has increased since charge is conserved but length is not:

So the field is:

6. [6 points] A long cylindrical wire of radius r = 3mm carries 100A of current. If the current density is uniform throughout the cross-section of the wire, what is the magnitude of the magnetic field at a radius of r = 1mm within the wire?

The current density is

Use Ampere’s Law to find the magnetic field by choosing a path C at a radius r = 1mm:

7. [8 points] An infinitely long insulated wire carrying a current I = 50 A is bent into a 270º arc (radians) of radius R=2 cm. The current comes in from infinitely far away from above, and exits to the left infinitely far away. Find the magnitude of the field B at the center of the arc. The wire begins and ends its turn at the locations indicated.

There are 3 components to the field: 2 semi-infinite segments of wire, and the ¾ circle. The magnitude of the field from a semi-infinite wire is half of that of an infinite wire (see homework, or look at direct integration of the Biot-Savart law):

in z direction.

From the circular wire, the field is:

The magnetic field from each section contributes in the same direction (z). Thus the total magnitude is:

8. A stationary neutral atom resides at the center of a Cartesian coordinate system and has a magnetic dipole moment of aligned in the direction.

(a) [6 points] What is the ratio of the magnitude of the magnetic field from the atomic dipole at y = 100 nm to that at y = 25 nm? (The radius of the atom is about 0.1 nm, where 1 nm = 10-9 m)

The field of a dipole at distances large compared to its size falls off with the cube power of distance:

So the ratio is:

(b) [6 points] If a magnetic field is present everywhere with the form , what is the magnitude of the acceleration of the dipole if the mass of the atom is 10-25 kg.

9. [6 points] The magnetic field of a large solenoid is used to keep a proton in a perfect circular orbit. The solenoid has 1000 windings per meter of length and has a radius of 1 m. If the proton has a velocity of , what is the minimum current needed to keep the proton orbiting within the confines of the solenoid in a plane perpendicular to the solenoid axis? The proton mass is and its charge is .

10. [6 points] A rod of length L = 0.5 m and mass m = 0.5 kg carries a current I = 20 A in the direction shown. The rod is aligned parallel to the z axis, and a uniform magnetic field is present: . The rod is suspended by two massless wires which bring the current to and away from the rod. The acceleration due to gravity is 10 m/s2 in the direction. What is the angle q that the suspension wires make with respect to the magnetic field direction?

11.  An oscillating LC circuit consists of a 0.002 H inductive coil and a 4 mF capacitor. The capacitor has a voltage drop of 0.75 V when the current through the coil is 0.03A.

(a)  [6 points] Find the maximum charge on the capacitor.

(b)  [6 points] Find the maximum current through the coil.

The total energy in the circuit is:

This must equal the energy when all charge is on the capacitor, or all current in the inductor:

12. [6 points] A circular conducting loop of wire increases in radius with time according to where is a constant. It is immersed in a constant magnetic field B perpendicular to the plane of the loop. What is the induced EMF in the loop?

13. A solenoid 21 cm long has a circular cross-section of 17 cm2. There are 210 turns of wire carrying a current of 2.5A.

(a)  [6 points] Find the magnitude of the magnetic field inside the solenoid assuming that it is essentially infinite in length (neglect end effects).

The magnetic field of a solenoid, obtained via Ampere’s Law for example, is:

(b)  [6 points] Find the total energy stored in the magnetic field inside the volume of the solenoid. Neglect end effects.

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