More Problems

Problem 1:

What current is required in the windings of a long solenoid that has 1000 turns uniformly distributed over a length of 0.4m in order to produce at the center of the solenoid a magnetic field of 10-4T?

Solution:

Concepts, principles, relations that apply to the problem:
The magnetic field inside a solenoid
Why do they apply?
We are asked to find the magnetic field in the center of a solenoid.
How do they apply?
Near the center of the solenoid B = m0nI = 4p10-7(1000/0.4)I T, with I in units of Ampere.
With B = 10-4 T, we need I = 10-4 A/(4p10-72500) = 31.8 mA.
(We have to assume that the length of the solenoid is much greater than its diameter, since not enough information is given for a more accurate calculation.)
Details of the calculation:
None

Problem 2:

What is the resistance of the following network?  Each ohmic resistor has resistance

Solution:

Concepts, principles, relations that apply to the problem:
Resistors in series and parallel
Why do they apply?
The circuit has enough symmetry so that we can analyze it like a simple circuit with resistors in series and parallel.
How do they apply?
 

Rtotal = (3/2)R.

Details of the calculation:
None

Problem 3:

A wire carries a steady current of 2.4 A.  A straight section of the wire is 0.75 m long and lies along the x-axis within a uniform field B = (1.6k) T.  If the current is in the positive x-direction, what is the magnetic force on the section of wire.

Solution:

Concepts, principles, relations that apply to the problem:
The Lorentz force on a wire,
Why do they apply?
We are asked to find the magnetic force on a section of wire.
How do they apply?
The force on the wire is given by F = IL´B.

The direction of L´B is the -j direction.  Since L and B are perpendicular to each other
F = ILB = (2.4 A)(0.75 m)(1.6 T) = 2.88 N.
The force on the section of wire is F = -2.88 Nj.

Details of the calculation:
None

Problem 4:

In the circuit below, let V1 be a 1V battery,  V2 be a 2V battery, V3 be a 3V battery, and let each resistor be a 1W resistor.  Pick directions for the currents as shown in the diagram.  Find I1, I2, and I3.

Solution:

Concepts, principles, relations that apply to the problem:
Kirchhoff's rules
Why do they apply?
We can find the the currents I1, I2, and I3 using Kirchhoff's rules.  The junction rule states that the sum of the currents entering a junction must equal the sum of the currents leaving that junction.  The loop rule states that the sum of the potential differences around any closed circuit loop must be zero.
How do they apply?
Junction A:  I1 + I3 = I2
Loop 1: V1 - I1R1 - I2R2 - V2 - I1R4 = 0
This yields
1V - I11W - I21W - 2V - I11W = 0,   or   2I1 + I2 = -1A.  (Units: V/W  = A)
Loop 2: V3 - I3R3 - I2R2 - V2 = 0
This yields
3V - I31W - I21W - 2V = 0,  or  I3 + I2 = 1A.
We now have to solve the three equations,
I1 + I3 = I2,
2I1 + I2 = -1A,
I3 + I2 = 1A,
for the three unknown currents.
Eliminate I3:  I3 = 1A - I2 using the third equation.  The first two equations the yield
I1 + 1A - I2 = I2,    I1 - 2I2 = -1A,
2I1 + I2 = -1A.
Eliminate I2:  I2 = -1A - 2I1.  We now can solve for I1.
I1 + 2A + 4I1 = -1A,    5I1 = -3A,   I1 = -(3/5)A.
Now we can solve for I2 and I3.
I2 = -1A - 2I1 = -1A + (6/5)A = (1/5)A
I3 = I2 - I1 = (4/5)A

I2 and I3 flow in the directions picked in the diagram, I1 flows in a direction opposite to the one picked in the diagram.

Details of the calculation:
None

Problem 5:

A large number, N, of closely spaced turns of fine wire are wound in a single layer upon the surface of a wooden sphere of radius R, with the planes of the turns perpendicular to the axis of the sphere and completely covering its surface.  If the current in the windings is I, determine the magnetic field at the center of the sphere.

Solution:

Concepts, principles, relations that apply to the problem:
The Biot-Savart law, the principle of superposition
Why do they apply?
A steady current is flowing in a circular loop and we are asked to find the magnetic field on the axis of this loop.
How do they apply?
(a) Consider a ring at q with width dq.  From the Biot-Savart law we have
dB(r) = (m0/4p)[Idl´(r-r')/|r-r'|3].  This yields the field at the origin due to the ring.
dB(r) = (m0/2)IR2sin2q/R3 k
In terms of a surface current density k we have I = kRdq and dB = (m0/2)ksin2qdqk.
The field due to all loops is 
.
We have Nd = pR, d = diameter of the wire, k = I/d = IN/(pR).
Therefore the field at the origin is B = (m0IN/(4R))k
Details of the calculation:
None

Problem 6:

A hydrogen atom consists of a proton and an electron separated by about 5 ´ 10-11 m.  If the electron moves around the proton in a circular orbit with a frequency of 1013 Hz, what is the magnetic field at the position of the proton due to the moving electron?

Solution:

Concepts, principles, relations that apply to the problem:
The Biot-Savart law, the magnetic field on the axis of a current loop
Why do they apply?
We treat the electron orbiting the proton as a current loop.  The current in the loop is the charge that passes a point per unit time, I = electron charge times orbiting frequency.
How do they apply?
Magnetic field at the center of a current loop:  B = (m0I/(2R)).
If electron orbits clockwise in the xy-plane, then the direction is positive z direction.
Here I = 1.6*10-19 C*1013/s = 1.6*10-6 A,  R = 5*10-11 m.
B = 4p*10-7*1.6*10-6/(10-10) T = 2*10-2 T.
Details of the calculation:
None