ELECTRIC POTENTIAL ENERGY

1. Electric current is the flow of electrical charge. If a small amount of charge flows in
time , then the current is . If the current is constant in time, then in time , the
curren
dq
dq i dq t
dt
=
t that flows is . The unit of charge is ampere, which is define as:
1 ampere 1 coulomb
second
A car's battery supplies upto 50 amperes w
q=i×t
=
3
hen starting the car, but often we need to deal
with smaller values:
1 milliampere 1 ma 10
1 mi
= = − A
6
9
12
croampere 1 10
1 nanoampere 1 n 10
1 picoampere 1 p 10
2. The direction of current
A A
A A
A A
μ −
−
−
= =
= =
= =
flow is the direction in which positive charges move. However, in
a typical wire, the positive charges are fixed to the atoms and it is really the negative
charges (electrons) that move. In that case the direction of current flow is reversed.
3. Current flows because something forces it around a circuit. That "something" is EMF,
electromotive force. But remember that we are using bad terminology and that EMF is not
a force - it is actually the difference in electric potentials between two parts of a circuit. So,
in the figure below, is the EMF which causes current to flow in the resistor.
How much current? Gener
a b V=V −V
ally, the larger is , the more current will flow and we expect
. In general this relation will not be completely accurate but when it holds, we say
Ohm's Law applies: . Here, i
V
I V
I V R V
R I
∝
= = s called the resistance.
+ −
conventional
current flow
electron flow
device
1 R 2 R
1 V 2 V
a
a b
1 b R
2 R
eq R
1 i
2 i
3 i
4. Be careful in understanding Ohm's Law. In general the current may depend upon the
applied voltage in a complicated way. Another way of saying this is that the resistance
may depend upon the current. Example: when current passes through a resistor, it gets
hot and its resistance increases. Only when the graph of current versus voltage is a
straight line does Ohm's Law hold. Else, we can only define the "incremental resistance".
1 2 3
5. Charge is always conserved, and therefore current is conserved as well. This means that
when a current splits into two currents the sum remains constant, i =i +i.
1 1 2 2 1 2 1 2
1 2
6. When resistors are put in series with each other, the same current flows through both. So,
and . The total potential drop across the pair is ( ).
. So resi eq
V iR V iR V V V iR R
R R R
= = = + = +
⇒ = + stors in series add up.
1 2
1 2
1 2
1 2
1 2
7. Resistors can also be put in parallel. This means that the same
voltage is across both. So the currents are , .
Since it follows that
1 1 1
eq
eq
V iV i V
R R
i i i i V V V
R R R
R R R
= =
= + = = +
⇒ = + 1 2
1 2
or .
This makes sense: with two possible paths the current will find
less resistance than if only one was present.
8. When current flows in a circuit work is done. Suppose a sma
eq
R R R
R R
=
+
ll amount of charge is
moved through a potential difference V. Then the work done is . Hence
dq
dW =Vdq=V idt
V (volts )
I(Amps)
obeys Ohm's Law
does not obey
Ohm's Law
R i
V
Δ
=
Δ
C
− + •
i
R
+
−
R
i
i
i
ε
b
d c
a
2 2
2
( because ). The rate of doing work, i.e. power, is .
This is an important formula. It can also be written as , or as . The unit of
power is: 1 volt-a
Vidt i Rdt iV PdW i R
R dt
P V P iV
R
= = = =
= =
mpere 1 joule coulomb 1 joule 1 watt.
coulomb second second
9. Kirchoff's Law: The sum of the potential differences encountered in moving around a
closed circuit is zero. This law is easy to prove: si
= ⋅ = =
nce the electric field is conservative,
therefore no work is done in taking a charge all around a circuit and putting it back
where it was. However, it is very useful in solving problems. As a trivial example,
consider the circuit below. The statement that, starting from any point a we get back
to the same potential after going around is: . This says 0. a a V −iR+ε=V −iR+ε=
10. We can apply Kirchoff's Law to a circuit that consists of
a resistor and capacitor in order to see how current flows
through it. Since , we can see that 0. Now
differentia
q VC q iR
C
= + =
0
te with respect to time to find 1 0,
or 1 . This equation has solution: . The
product is called the time constant , and it gives the
time by which the c
t
RC
dq di R
C dt dt
di i
dt RC
RC
i ie
τ
−
+ =
= − =
urrent has fallen to 1/ 1/ 2.7 of the
initial value.
e ≈
2 3 4
2 3
A reminder about the exponential function, 1
2! 3! 4!
From this, d 12 3 4 Similarly, d
dx 2! 3! 4! dx
x
x x x x
e x x x x
e x x x e e− e−
≡ + + + + + ⋅ ⋅ ⋅ ⋅
= + + + + ⋅⋅⋅ = = −
11. Circuits often have two or more loops. To find the voltages and currents in such situations,
it is best to apply Kirchoff's Law. In the figure below, you see that there are 3 loops and
you can see that:
2 R
i
1 i 2 i
1 ε
2 ε
3 i 3 R 1 R
I
V
L
A I
1 3 2
1 1 1 3 3
3 3 2 2 2
0
0
i i i
i R i R
i R i R
ε
ε
+ =
− + =
− − + =
( ) ( ) 1 2 3 2 3 1 3 2 1 3 1 2 2 1
1 2 3
1 2 2 3 1 3 1 2 2 3 1 3 1 2 2 3 1 3
Make sure that you understand each of these, and then check that the solution is:
12.
i R R R i R R R i R R
R R R R R R R R R R R R R R R R R R
ε + −ε ε −ε + −ε −ε
= = =
+ + + + + +
A charge inside a wire moves under the influence of the applied electric field and suffers
many collisions that cause it to move on a highly irregular, jagged path as shown below.
Nevertheless, it moves on the average to the right at the "drift velocity" (or speed).
13. Consider a wire through which charge is flowing. Suppose that the number of charges
per unit volum
( )
d
e is . If we multiply by the crossectional area of the wire and the
length , then the charge in this section of the wire is . If the drift velocity
of the charges is v , then t
n n A
L q= nAL e
d
d d
he time taken for the charge to move through the wire is
. hence the current is v . From this we can calculate the drift
v /v
velocity of the charges in terms of the measur
tL iqnALe nAe
t L
= = = =
d
d
ed current, v . The current density,
which is the current per unit crossectional area is defined as = v . If varies
inside a volume, then we can easily generalize and write,
i
nAe
j i ne j
A
=
=
i= ∫j⋅dA.
􀁇 􀁇
q
v 􀁇
B 􀁇
θ
F =qv×B
􀁇 􀁇 􀁇
+ + + + + +
− − − − − −
+
v B F = q B
E F = qE
⊗
⊗
⊗
⊗
qE qvB v E velocity selector !!
B
= ⇒ =
⊗
⊗
⊗
⊗
⊗
⊗
⊗
⊗
⊗
⊗
⊗
⊗
+ +