Circuits Kirchhoff’s Rules

Visualizing Circuits as a Gravity Problem In electricity potential (voltage) is synonymous with height in mechanics. This battery lifts charges to a potential (height) of 12 V. The charges return to the battery in the circuit. 12 V 4 Ω 0 V Voltage is positive for the battery. Voltage is negative in the circuit.

Types of Circuits Series Parallel More than one path One path

Kirchhoff’s Loop Rule The potential difference in a loop of a circuit is zero. The voltage of all the elements of a circuit must add to zero. Based on conservation of energy. If you start at a point, move in any direction or in any manner, and eventually return to the starting point there can be no change in energy or potential. Whenever you return to the starting location final minus initial will always be zero.

Kirchhoff’s Loop Rule In order move and eventually return to the starting conditions an object must move both positively and negatively. Voltage is analogous to height in gravity. Moving upward increases potential and potential energy. Batteries and generators act like charge escalators lifting charges against the natural tendency to fall, and have positive potential. Moving downward decreases potential and potential energy. Electrical components use the energy created by batteries and generators. As the charges fall through the circuit they lose energy to components such as resistors, which have negative potential.

Potential of Batteries in Loops The potential of devices that create a potential difference, such as batteries and generators, is known as electromotive force, emf ( ε ) While batteries are associated with a positive potential difference (emf) there are times when a battery may be labeled as having a negative potential difference (emf). If a battery is placed in a circuit backwards, then its potential difference (emf) is reversed. When there are two or more batteries the positive potential difference (+ε ) is assigned to the strongest battery, and the negative potential difference (−ε )is assigned to any reversed weaker batteries.

Example 1 Solve for the potential difference across resistor R . R 1 V 3 V 6 V Start with the emf of the strongest battery and trace the circuit from the positive terminal of the strongest battery to the negative terminal of the strongest battery.

Example 2 Solve for the potential difference across resistor R . R 2 V 4 V 12 V Start with the emf of the strongest battery and trace the circuit from the positive terminal of the strongest battery to the negative terminal of the strongest battery. The second smaller battery is oriented in the same direction (+emf) as the stronger battery. These two batteries work together acting as one larger battery.

Example 3 Solve for the potential difference across resistor R . R 2 V 4 V 12 V Start with the emf of the strongest battery and trace the circuit from the positive terminal of the strongest battery to the negative terminal of the strongest battery. The second smaller battery is oriented opposite (−emf) to the stronger battery. These two batteries oppose each other, but the stronger battery wins and sets the overall current direction.

Example 4 R 12 V R 12 V This circuit has two loops. Solve for the potential difference across resistor R . This circuit has two loops. We are only interested in the loop containing resistor R . R 12 V R 12 V The second unlabeled resistor must also have a potential of 12 V, since it and the battery are the only two elements in the smaller loop.

Kirchhoff’s Junction Rule The current entering a circuit junction must equal the current leaving the junction. Based on conservation of charge. Since charges cannot be created or destroyed the number of charges entering and leaving must be equal. Charges cannot appear or disappear. The direction of positive current is from the positive terminal of a battery to the negative terminal of a battery.

Example 5 Solve for the current passing through resistor R . R 8 A The battery is the source of the current and the direction of the current is from the positive terminal of the battery to the negative terminal. As you trace the circuit from terminal to terminal the current has to split up when it reach a junction. This series circuit has no junctions. The current must be the same everywhere. The current in the two unlabeled resistors must also be 8 A .

Example 6 Solve for the current passing through resistor R . 6 A 2A The direction of the battery sets the current direction. Examine the direction of the current due to the battery(s) at the top junction. R 2 A 6 A 8 A 6 A 2A We can check the bottom junction as a double check.

Example 7 Solve for the current passing through resistor R . 9 A 3 A 6A The direction of the battery sets the current direction. Examine the direction of the current due to the battery at the top junction. R 9 A 3 A

Example 8 Solve for the current passing through resistor R . 9 A 3 A 6A The direction of the battery sets the current direction. Examine the direction of the current due to the battery at the top junction. R 9 A 3 A

Implications of Kirchhoff’s Rules Series Parallel Kirchhoff’s Rules Voltage Adds Same Voltage in a loop containing a power source is zero. Conservation of energy. The battery lifts charges to high voltage and energy. Then the charges move in the circuit dropping the same voltage to return to the battery. The positive lift and negative drop must cancel exactly. Current The current entering a junction must equal the current leaving a junction. Conservation of charge. If there are no junctions (series), then current must be the same everywhere. If current reaches a junction it splits up, but must still total to the amount that entered the junction.

Example 9 Loop Rule R1 8 A 6 V __ V __ A 3 V 1 V R2 R3 2 8 Solve for the missing voltages and currents Loop Rule R1 8 A 6 V __ V __ A 3 V 1 V R2 R3 2 8 Junction Rule No Junctions. Current must be the same.

Example 10 Loop Rule __ A __ V 9 V 1 A 3 A R1 R2 9 9 2 Junction Rule Solve for the missing voltages and currents Loop Rule __ A __ V 9 V 1 A 3 A R1 R2 9 9 2 Junction Rule

Equivalent Resistance The total resistance of two or more resistors in a part of a circuit or in the entire circuit. The value of one resistor that is equal to, and thus can replace, all the resistors you are evaluating. Series Parallel

Example 11 Determine the equivalent resistance for the portion of the circuits shown. 2 Ω 4 Ω Example 12 2 Ω 4 Ω

Equivalent Capacitance When two or more capacitors are wired together the capacitance adds to form a sum of capacitance, or equivalent capacitance. The equivalent capacitance is the value of the capacitor that would be needed to do the same job as several capacitors added together. Adding Capacitors in Series Adding Capacitors in Parallel

Example 13 Solve for the equivalent capacitance C1 = 2 F C2 = 4 F There is only one path from left to right: Series Be careful: we just solved for 1/C . We are looking for C .

Example 14 Solve for the equivalent capacitance C1 = 2 F C2 = 4 F There are two paths from left to right: Parallel

Example 15 Solve for the equivalent capacitance This is parallel and series Turn the parallel part of the circuit into one capacitor 20 μF 10 μF Now add these capacitors in series

Multiple Batteries Batteries supply the potential that drives the entire circuit. Potential adds in series. When multiple batteries are placed in series they simply add. However, if a battery is reversed it acts as though it is negative. V1 = 6 V V2 = 2 V 6 + 2 = 8 V V1 = 6 V V2 = −2 V 6 - 2 = 4 V