CIRCUITS

EMF

An emf device performs work on charges in order to maintain a potential difference between its output terminals. In the equation below, we have the work per unit charge of a device, where dW is the work the device does to force positive charge dq from the negative to the positive terminal.
1.bmp(1)
E is representative of the emf (work per unit charge) of the device.

Typically this "work per unit charge" is measured in volts, as is the potential difference. An ideal emf device is one that lacks any internal resistance. The potential difference between its terminals is equal to the emf. A real emf device has internal resistance. The potential difference between its terminals is equal to the emf only if there is no current through the device.

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A simple electric circuit, in which a device of emf E does work on the charge carriers and maintains a steady current i in a resistor of resistance R.



Rules for Analyzing Circuits


Resistance Rule:

The change in potential in traversing a resistance R in the direction of the current is -iR; in the opposite direction it is +iR.

EMF Rule:

The change in potential in traversing an ideal emf device in the direction of the emf arrow is +E; in the opposite direction it is -E.

Loop Rule, or Kirchhoff's Voltage Law:

The algebraic sum of the changes in potential encountered in a complete traversal of any loop of a circuit must be zero.
This law is also called Kirchhoff's second law, Kirchhoff's loop (or mesh) rule, and Kirchhoff's second rule:

"The sum of the emf's in a closed circuit is equal to the sum of potential drops in that same circuit".

Similarly to KCL, it can be stated as:
external image 08d7bd7060be987d4da37b7fc263a740.png
where n is the total number of voltages measured. The voltages may also be complex:
external image a926265ee2e2ef9f177b3ded6f80fc2c.png
This law is based on the conservation of energy, whereby voltage is defined as the energy per unit charge. The total amount of energy gained per unit charge must equal the amount of energy lost per unit charge. This seems to be true as the conservation of energy states that energy cannot be created or destroyed; it can only be transformed into one form to another.

Junction Rule:

The sum of the currents entering any junction must be equal to the sum of the currents leaving that junction.



Single-Loop Circuits

The current in a single-loop circuit containing a single resistance R and an emf device with emf E and internal resistance r is:
2.bmp(2)which reduces to i=E/R for an ideal emf device with r = 0.




Power

When a real battery of emf E and internal resistance r does work on the charge carriers in a current i through a battery, the rate P of energy transfer to the charge carriers is:
3.bmpwhere V is the potential across the terminals of the battery. The rate P(r) at which energy is dissipated as thermal energy in the battery is:
4.bmpThe rate P(emf) at which the chemical energy in the battery changes is:
5.bmp(3)


Resistors
Resistors come in various shapes and sizes and their main use is to act as a buffer in an electric circuit. When a resistor is placed in a circuit, a voltage drop occurs across the resistor's terminals that is proportional to the current through the use of Ohm's law.

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Various Resistors
Resistors come in a variety of different resistances (Ω) and a color-coded system was invented to make it easier to determine what resistance a resistor has. This system is useful to know when building or analyzing circuits and when trying to solve equations when the resistance is not immediately clear or given.
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Resistor Color Code Chart

Resistors in Series and in Parallel

Resistors can be wired in series and in parallel throughout a circuit. They can also be replace by a group of resistors with an equivalent resistance.

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(a) Resistors in series.

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(b) Resistors in parallel.


(a) Calculating Equivalent Series Resistance
When resistors are in series, they have the same current and the sum of potential differences across the resistances is equal to the applied difference V. The equivalent resistance that can replace a combination of n resistances in series is:
6.bmp(4)
(b) Calculating Equivalent Parallel Resistance
When resistors are in parallel, they have the same potential difference V and the same total curent i as the individual resistances. The equivalent resistance that can replace a combination of n resistances in parallel is:
7.bmp(5)



RC Circuits

A resistor–capacitor circuit (RC circuit), is an electric circuit composed of resistors and capacitors driven by a voltage or current source. The 1st order RC circuit composed of one resistor and one capacitor, is the simplest example of an RC circuit.
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Simple RC Circuit

Series RC circuit.
Series RC circuit.

Parallel RC circuit.
Parallel RC circuit.

Charging a Capacitor
When an emf E is applied to a resistance R and a capacitance C in series, the charge on the capacitor increases according to:
8.bmp(6)in which CE=qo is the equilibrium (final) charge and RC=t is the capacitive time constant of the circuit. During the charging, the current is:
9.bmp(7)


How not to charge a capacitor.

Discharging a Capacitor
When a capacitor discharges through a resistance R, the charge on the capacitor decays according to:
10.bmp(8)During the discharging, the current is:
11.bmp(9)


A capacitor being discharged.
Derivation
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Example 1:
When resistors 1 and 2 are connected in series, the equivalent resistance is 16.0 Ω. When they are connected in parallel, the equivalent resistance is 3.0 Ω. What are (a) the smaller resistance and (b) the larger resistance of these two resistors?
cur2.bmp

Example 2:
Four 18.0 Ω resistors are connected in parallel across a 25.0 V battery. What is the current through the battery?
Cur1.bmp