Work, Energy and EMF

The figure shows an emf device (battery) that is part
of simple circuit. The device keeps one terminal (+) at a higher electric
potential than the other terminal (-). We represent the emf of the device
with an arrow that points from the negative to positive terminal. This
is the direction in which the device produces positive charge carriers 
to move through itself. The device also produces a current around the
circuit in the same direction.

In any time interval dt, a charge dq passes through any cross-section of this circuit such as aa. The
same charge must enter the emf device at its low potential end and leave
at its high potential end. This device must do amount of work dW on the charge dq to force it to move in
this way. We define the emf of the emf device in terms of this work

The emf of emf device is the work per unit charge that the device
does in moving the charge from its low potential terminal to high potential
terminal. The unit is J/C which is the Volt

Work, Energy and EMF

An emf device performs two functions: it maintains
a potential difference and it moves charge from one terminal to the
other inside the device

An ideal emf device lacks any internal resistance to the internal movement
of charges from terminal to terminal. For example, an ideal battery
with an emf of 12V always has a voltage of 12V between its terminals

A real emf
device, such as real battery, has internal resistance to the internal
movement of charge. When a real emf device is not connected to circuit,
the potential difference between its terminals is equal to its emf.
But when it has current through it, the potential difference between
its terminal, is different that the emf.  

Work, Energy and EMF

An emf device does positive work on the charge passing through it
if the current is from the negative to positive terminal and negative
work if the current is in the other direction

Positive work done by an emf device results in a decrease
in the store of energy of the device, chemical energy in the case of
a battery. Negative work results in an increase in the store of energy
of the device. If the device is a battery, then in the first case it
is discharging and in the second it is charging.

The potential difference across an ideal emf device
does not change with direction of current through the device.

Calculating the Current in a Single Loop

There are two ways to calculate the current in the simple single-loop
circuit; one method is based on energy conservation and the other is
based on potential. The circuit consist of an ideal battery with emf , a
resistor R, and two connecting wires (zero resistance).

Energy method

The equation,  P=i<sup>2 R, tells us that in a time interval
dt an amount of energy given by 
i2 R will appear in the resistor as thermal energy. During the same
interval, a charge dq = idt will have move the battery, and the battery will have done work
on this charge, according to the definition of
emf

,
equal
to

Form the principle of conservation of energy, the work done by
the battery must equal the thermal energy in the resistor

Energy method

Emf is the energy per unit charge transferred to the moving charges by
the battery. The quantity iR is the energy per unit charge transferred from the moving charge
to thermal energy within the resistor. The energy per unit charge transferred
to the moving charges is equal to the energy per charge transferred
from them. Solving for I, we find

Potential Method

Suppose we start at any point in the circuit to the right and mentally
proceed around the circuit, adding algebraically the potential difference
that we encounter. When we arrive at our starting point, we must have
returned to our starting potential. This is called the loop rule in
circuits

This is often referred to as Kirchoffs loop rule
or voltage law (Kirchoff is a german physicist).

Potential Method

Let us starting a point a, whose potential is Va, and mentally walk clockwise around the circuit until we are back
at a, keeping track of the voltages along the way. Our starting point
is at the low-potential terminal of the battery. Since the battery is
ideal, the potential difference between its terminals is zero. So when
we pass through the battery to high potential terminal, the change in
potential is +.

As we walk along the top wire to the top end of the resistor, there
is not potential change because the wire has negligible resistance so
it is same as the high-potential terminal of the battery . When we pass
through the resistor, however the change in potential is iR.

We return to point a along the bottom wire. Again, since the wire has negligible resistance,
we again find no potential change. Because we transverse a complete
loop, our initial potential, as modified for potential changes along
the way, must equal to our final potential; that is

Since Va cancels we can

Potential Method

To prepare for circuits of greater complexity, let us lay down two
global rules for finding potential differences as we move around the
a loop:

(a) rightward

(b) All the same

(c