- 15PHY 132
Section 5:
h
Electrical power, P:
- Several relationships involving power and energy.
Ex. 5-1: Find: a) the resistance of a 55 W, 120 V light bulb.
b) how many of these bulbs can be lit at once on a circuit with a 20 A fuse.
The circuit's "pump" (battery or generator) is called a source of EMF, electromotive "force".
Definition of EMF, E
(For example, with a 1.5 V flashlight battery, E is the 1.5 V.)
Kirchhoff's Laws:
-Loop Rule. Follows from conservation of energy.
-Point Rule. (Or “Junction Rule”) Follows from conservation of charge.
(Current, like water, flows "downhill.")
Ex. 5-2: A 10 Ω resistor is placed in series with a 20 Ω resistor across a 1.5 V emf. Find the current.
(Notice from the solution that resistors in series are equivalent to a single resistor of resistance
Req = R1 + R2 + R3 + ...)
Ex. 5-3: 100 mA flows when 10 Ω is placed across an old 1 1/2 V battery. Find:
a.
its internal resistance,
b. the terminal potential.
E usually ≠ "voltage" between battery's terminals due to
internal resistance:
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(Battery goes dead when r gets large.)
Ex. 5-4: A 10 Ω resistor is placed in parallel with a 20 Ω resistor across a 1.5 V emf. Find the
current through the battery.
(Notice from the solution that resistors in parallel are equivalent to a single resistor of resistance
1/Req = 1/R1 + 1/R2 + 1/R3 + ...)
Ex. 5-5: Find the current through each battery.
Now, you do this one:
Ex. 5-6: What does the ammeter read?
Ans: .125 A
Electrical safety:
Too much current through a wire heats it too much. Avoided by using a fuse or circuit breaker.
Electricity flowing through the body can cause two kinds of injury:
-burns
-interference with nerve impulses to heart or lungs.
(Injury depends on amount of current flowing through body.)
PRECAUTIONS:
-Many devices are grounded. (A ground wire goes to a long stake in the ground, or sometimes a
water pipe. If something goes wrong, a connection to ground gives electricity someplace to go
that isn't through you.)
- Be careful you're not grounded. (It won't flow through you with nowhere to flow to.)
- Keep one hand in your pocket when working on a live circuit (to keep current out of your chest.)
- Be very careful touching someone else being shocked.
- 17Section 6:
Field of a continuously distributed charge: Can use superposition, like in section 1. The only
difference is that there, you added the fields of a finite number of separate charges; here, it's an
infinite number of charges, all joined together.
To find E using superposition: Consider the object to be an infinite number of point charges, joined
together.
Ex. 6-1: A charge of +Q is uniformly distributed along the
rod. Find E at the origin.
Procedure (for a charged line or curve):
1. Divide it into pointlike pieces, and write down field of one piece.
2. Use charge density to substitute for dq.
(λ = q/l, σ = q/A, ρ = q/V)
3. Other substitutions, if necessary, to get it into an integrable form.
4. Integrate. (Superpose contributions from all pieces.)
Remember to treat
as a vector:
Ex. 6-2: A charge of +Q is uniformly distributed
along the rod, which
 is bent into a quarter of a circle
of radius r. Find E at the origin, the center of the
circle.
- 18Ex. 6-3:
A charged rod is bent into the spiral shape shown. Its charge is distributed in a non-uniform way,
given by λ = rλ0, where λ0 is a constant. Find E at the origin. (Hint: use θ as the variable of
integration.)
You work on this one:
Ex. 6-4: A rod of length 2a lies along the y axis as
shown. It has the same charge per unit length, λ,
everywhere. Find the electric field vector a distance a to
the right of its center.
Hints:
1. Your variable of integration, θ, runs from -π/4 to
π/4.
2. It can be shown that the length of a small piece of
2
the rod is dy = (r dθ)/a.
3. Because of the symmetry of the situation, the y
components from the top half of the rod cancel those
from the bottom half.
Ans:
√
̂
- 19Section 7: Magnetic Forces
There are two kinds of magnetic poles, called north and south. (N and S are to magnetism what +
and - are to electricity.) Opposites attract, likes repel.
The magnetic field vector, ⃑ , points in the direction of the force on a N pole, just as the electric field
vector, ⃑ , points in the direction of the force on a + charge.
One difference between electricity and magnetism is that there are no magnetic monopoles. (You
can have a proton without a nearby electron, but you can't have a N without a nearby S.) So, let's
restate B's direction in terms of a dipole:
A compass is a little magnet, mounted on a pivot. It will feel a torque in the field of
another magnet.
(The end which points toward the north in Earth's field is labeled N.)
The direction of the field is the direction a compass would point. (Field lines
outside a magnet run N to S.)
Magnetic force on an electric charge:
A stationary charge feels no force:
A moving charge does feel a force, of magnitude F = q v B sinθ.
So, you could define B as B = F
_
q v sinθ
units: N = tesla
c(m/s)
4
10 gauss = 1 tesla
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

Direction of this force (on a + charge): Perpendicular to both v and B , according to the right hand
rule.
  
F  qv  B
examples:
n̂ = unit normal vector. (Wrap fingers around origin, pointing from first vector toward second.
Thumb points in direction of n̂ .)
iˆ
 
Cross product: A  C = A C sinθ n̂ = Ax
Cx
ˆj
Ay
Cy
kˆ
Az
Cz
(determinant)
(I will run through how to evaluate this on the board.)
7
Ex. 7-1: The beam of electrons in a CRT has a velocity of (1.60 iˆ + 2.90 ĵ ) x 10 m/s. Earth's field
here is ⃑ = 30.0 ĵ - 50.0 k̂ μT. Find the force on an electron in the beam.
You work on this for a few minutes. ( means vector pointing toward you, x means it's pointing
away from you.)
Ex. 7-2:
In each case, state
whether there is a force on the
charged particle. If there is, say
whether its direction is +x, -x, +y, y, +z or -z.
- 21-
If charges move perpendicular to the field, they follow a circular path.
(I will write out how to determine the radius of the circle.)
Lorentz force equation.
Ex. 7-3: Find the electric field which allows the proton to keep a
constant velocity.
Charges moving through a wire, instead of empty
space, feel this same force. So, there is a force on a
current carrying wire.
Q = total free charge in wire.
Equation for the force on a current-carrying wire.
Definition of length vector, ⃑
Ex. 7-4: Is there a force on the wire? If
there is, say whether its direction is +x, -x,
+y, -y, +z or -z.
Ex. 7-5: A power line carries 50 A in the +x direction. Earth's field (.6 gauss) points down the z
axis. Find the magnetic force on a section of line 65 m long.
- 22Torque on a current loop:

- Definition of magnetic moment, 
Area vector:
magnitude = A
direction: perpendicular to plane of loop, by right hand rule.
Ex. 7-6: The coil has 300 turns, an effective radius of 7 cm, and carries
2.5 A. a) How large is the torque? b) Is it clockwise or
counterclockwise?
You work on this:
Ex. 7-7: Is there a torque on the loop? If
there is, say whether its direction is +x, x, +y, -y, +z or -z.
- 23So, what is magnetism really?
Special relativity (Phy 133, sec 5): Since the speed of light is the same to all observers, distance and
time must be relative. A moving clock runs slower, and is shortened along its direction of motion.
Consider two charges, at rest in the primed frame of
reference:
In the frame where they are at rest, the force between them is as described in
section 1:
Looking from the
 other frame of reference changes the
angles so that F does not point directly away from the
other charge. (The change in x is not the only factor, but
contributes.)
Break the force into non-perpendicular components, one pointing away from the other charge

(called electric) and one perpendicular to v (called magnetic).
Example: (Uncharged) current-carrying wires attract. FE from electrons is cancelled out by FE from
surrounding metal ions. But, FM from moving electrons is not cancelled by unmoving metal ions.
- 24Section 8:
I will demonstrate that magnetic field lines around a
long straight wire are circles.
Which way they go around is given by the right
hand rule: Point thumb along the current, fingers
wrap around like the field lines.
Ampere's law
To find B:
1. Decide what the field looks like. (Right hand rule, etc.)
2. Pick a path which the field point lies on, and which has the same symmetry as the field.
3. Evaluate
along the path.
4. Set it equal to μ0I.
5. Substitutions, solve for B.
Ex. 8-1: Find B a distance r from a long straight wire.
example: B points into the page at P1 and out at P2.
Field of a coil (or a loop of wire): Grab
coil with fingers wrapped around in
direction current goes. Thumb points in
direction of field inside.
- 25-
Solenoid = a long coil.
Field outside a solenoid is quite weak. (B  0) Field inside is uniform.
Ex. 8-2: Find B inside an infinitely long solenoid with n turns per unit length.
ans: path for Ampere's law:
Notice from the answer to that question that the field of a finite coil with N turns and length L is B 
(μ0IN)/L, as on formula sheet.
Also on formula sheet:
- B of a flat (i.e., short) coil.
- B of a toroidal (doughnut shaped) coil.
Ex. 8-3: A beam of ions has a circular cross section with a radius of 5.00
5
cm. The magnitude of its current density is J = 3.00 X 10 A/m2. Find the
magnitude of the magnetic field at points A, B and C.
Now, you try this: Ex. 8-4: A very long empty metal pipe with very thin walls carries a current of
5.0 A. Its radius is 1.0 cm. Find B at (a) r = .75 cm and at (b) r = 1.25 cm. (r = distance from center
of pipe.)
Ex. 8-5: Find B at point P.
Ex. 8-6: Find the force on wire D due to C, if each is 1.0 m long.
- 26-
Ex. 8-7: An electron inside the solenoid is going left at 2.3 x
7
10 m/s. Find the force on it.
You work on this:
Ex. 8-8 In each case, state whether there is a
force on the charged particle. If there is, say
whether its direction is +x, -x, +y, -y, +z or -z.
Is there a force on the straight wire? If
there is, say whether its direction is +x, -x,
+y, -y, +z or -z.
- 27Review for Exam 2:
1. How many watts are consumed by the 6.00 Ω resistor?
Ans: 13.5 W
2. The square loop of wire has an area of .0400 m2 and
carries 25.0 A as shown. A .500 T magnetic field is parallel
to the xy plane, 37 above the horizontal. What is the
torque vector acting on this loop?
Ans: –.261 ̂ + .346 ̂ – .150 ̂ N·m
3. A charged rod is bent to form a semicircle of radius .640 m.
It has a uniform charge per unit length of λ = 3.00 nC/m. What
is ⃑ , the electric field vector, at the center?
Ans: 84.3 ̂ V/m
4. A long straight wire has an outside radius of R, and
carries a current of I which has the same current density
everywhere. (Same current per unit area.) Use
Ampere's law to find an expression for B, the magnetic
field’s magnitude, at a point inside the wire a distance r
from the center. Express your answer in terms of r, R, I and fundamental constants.
Ans:
5. Short answer, 5 points each:
a. Which resistor consumes more power, or are
they the same?
- 28-
b. The charge on both rods is the same, but that charge is
distributed differently. Is the electric field at the origin
stronger in the case where the charge is distributed
uniformly, in the case where it is distributed according to λ
= 1/x, or is it the same in both cases?
c. A sheet of aluminum foil carries an evenly distributed
current out of the page as shown. What is the direction
of the magnetic field at point P?
d. Two circular loops of wire each carry 1.0 A. One is larger than the other. Which loop has a
stronger magnetic field at its center, or are they the same?
e. A proton moving north through a magnetic field feels a force toward the east. What is the
direction of the magnetic field?