1) A circuit requires a source of electromotive force (emf) to maintain current flow. Batteries act as emf sources by using chemical reactions to pump electrons in one direction, increasing their potential energy. 2) In a circuit with a battery and resistor, the battery's terminal voltage is lower than its emf due to the battery's internal resistance. Terminal voltage equals emf minus the voltage drop across the internal resistance due to current flow. 3) For resistors in series, the total resistance is equal to the sum of the individual resistances. Current flow is the same through all elements, so the total potential difference across the combination of resistors is equal to the sum of the individual potential

1. CIRCUITS

2. DIRECT-CURRENT PRINCIPLES
 A current is maintained in a closed circuit by a source of emf
 Source of emf
Any device that increase the potential energy of the circulating
charges
Can be thought of as a charge pump that forces electrons to move in
a direction opposite of the electrostatic field inside the source
Emf of a source
Work done per unit charge

3.  Consider the circuit consisting of
a battery connected to a resistor
Assume that the connection wires
have no resistance
Dropping the internal resistance of
the battery, the potential drop
across the battery equals the emf of
the battery
Because a real battery always has
some internal resistance r, the
terminal voltage is not equal to the
emf

4.  The battery, represented by the dashed
rectangle, consists of a source of emfε in
series with an internal resistance r
 Imagine a positive charge moving through
the battery from a to b in the figure
 As the charge passes from negative to the
positive terminal of the battery, the potential of
the charge increases by ε
 As the charge moves through the resistance r,
however, its potential decreases by the amount
Ir, where I is the current in the circuit

5.  The terminal voltage of the battery, ∆𝑉 = 𝑉𝑏 − 𝑉
𝑎, is therefore given
by ∆𝑉 = ε − 𝐼𝑟
from this expression , ε is equal to the terminal voltage when the
current is zero, called the open-circuit voltage
 The terminal voltage ∆𝑉 in the figure must be equal the potential
difference across the external resistance R , often called the load
resistance. That is, ∆𝑉=IR
Combining the two ε = IR + Ir
Solving for the current I=
𝜀
𝑅+𝑟

6.  The equation shows that the current in the simple circuit
depends on both the resistance external to the battery and the
internal resistance of the battery. Multiplying the previous
equation by I, we have
𝐼ε = 𝐼2
𝑅 + 𝐼2
𝑟
Which tells us that the total power output 𝐼ε of the source of emf
is converted at the rate of 𝐼2
𝑅 at which energy is delivered to the
load resistance, plus the rate 𝐼2
𝑟 at which energy is delivered to
the internal resistance if 𝑟 << 𝑅, most of the powered delivered
by the battery is transferred to the load resistance

7. RESISTORS IN SERIES
 The figure above shows how two or more resistors are connected in
series
 Here the current is the same in the two resistors, because any
change that flows through 𝑅1must also flow through 𝑅2

8.  Because the potential difference between a and b
equals 𝐼𝑅1 and the potential difference between b and c equals 𝐼𝑅2,
the potential difference between a and c is
∆𝑉 = 𝐼𝑅1 + 𝐼𝑅2= 𝐼(𝑅1 + 𝑅2)
 Regardless of how many resistors are in series, the sum of the
potential difference across the resistors is equal to the total
potential difference across the combination
 Applying Ohm’s Law to the equivalent resistor, we have
∆𝑉 = 𝐼𝑅𝑒𝑞
Equating the preceding two expressions, we have
𝐼𝑅𝑒𝑞 = 𝐼(𝑅1 + 𝑅2)
or 𝑅𝑒𝑞 = 𝑅1 + 𝑅2 (series combination)

9.  Analysis shows that the equivalent resistance of three or
more resistors connected in series is
𝑅𝑒𝑞 = 𝑅1 + 𝑅2 + 𝑅3 + …
Thus, the equivalent resistance of a series combination of resistors
is the algebraic sum of individual resistances and is always greater
than any individual resistance.