Introductory Physics Volume Two

60 Circuits 3.4
be an electric potential difference ∆V = V a − V b = − ∫ a
E ⃗ · ⃗dr between
b
the two terminals. It can be shown that, regardless of the shape of
the conductors, this electric potential difference is proportional to the
current if the material is ohmic.
Theorem: Ohm’s Law: Resistance
∆V = IR
Where R is called the resistance of the element.
The unit of resistance is the ohm which is one volts per amp. The
ohm is abbreviated as Ω, so that 1Ω = 1V/1A.
⊲ Problem 3.2
You have a wire with cross sectional area A and length L. Show that if
the terminals are placed at the ends of this conductor that the resistance
of this element is R = ρ L A .
⊲ Problem 3.3
You have a block of carbon, with sides of length a, 2a, and 3a. If
terminals are placed on two parallel sides we can make a resistor with
this block. We have three choices for the placement of the terminals,
the sides that are a apart, 2a apart or 3a apart.
(a) Which choice will produce the most resistance.
(b) Which choice will produce the least resistance.
§ 3.4 Electric Power
Suppose that you have a circuit element with two terminals, that
has a current I running through it and a potential difference ∆V between
the terminals. In a time dt an amount of charge dq = I dt will
pass through the element. All of that charge falls through the electric
potential difference of ∆V so that the charge dq looses an amount of
potential energy
dU = dq ∆V = I dt ∆V −→ dU
dt = I ∆V.
So we see that the element dissipates a power P = I ∆V .
Theorem: Electrical Power
The power dissipated in a circuit element is equal to the product
of the current through the element and the potential difference
between the terminals of the element.
P = I ∆V