Compound Direct Current Circuits
Describe the equivalent resistance of multiple resistors connected in a circuit.
Circuit elements may be connected in series and/or in parallel.
A series connection is one in which any charge passing through one circuit element must proceed through all elements in that connection and has no other path available. The current in each element in series must be the same.
A parallel connection is one in which charges may pass through one of two or more paths. Across each path, the potential difference is the same.
A collection of resistors in a circuit may be analyzed as though it were a single resistor with an equivalent resistance \(R_eq\).
The equivalent resistance of a set of resistors in series is the sum of the individual resistances. \[\begin{equation} R_{eq,s} = \sum_i R_i \end{equation}\]
The inverse of the equivalent resistance of a set of resistors connected in parallel is equal to the sum of the inverses of the individual resistances. \[\begin{equation} \dfrac{1}{R_{eq,p}} = \sum_i \dfrac{1}{R_i} \end{equation}\]
When resistors are connected in parallel, the number of paths available to charges increases, and the equivalent resistance of the group of resistors decreases.
Describe a circuit with resistive wires and a battery with internal resistance.
Ideal batteries have negligible internal resistance. Ideal wires have negligible resistance.
The resistance of wires that are good conductors may normally be neglected, because their resistance is much smaller than that of other elements of a circuit.
The resistance of wires may only be neglected if the circuit contains other elements that do have resistance.
The potential difference a battery would supply if it were ideal is the potential difference measured across the terminals when there is no current in the battery and is sometimes referred to as its electromotive force (emf, \(\mathcal{E}\)).
The internal resistance of a nonideal battery may be treated as the resistance of a resistor in series with an ideal battery and the remainder of the circuit.
When there is a current in a nonideal battery with internal resistance \(r\), the potential difference across the terminals of the battery is reduced relative to the potential difference when there is no current in the battery. \[\begin{equation} \Delta V_{terminal} = \mathcal{E} - I r \end{equation}\]
Measurement of current and voltage in a circuit
Describe the measurement of current and potential difference in a circuit.
Ammeters are used to measure current at a specific point in a circuit. Ammeters must be connected in series with the element in which current is being measured. Ideal ammeters have zero resistance so that they do not affect the current within the element that they are in series with.
Voltmeters are used to measure electric potential difference between two points in a circuit. Voltmeters must be connected in parallel with the element across which potential difference is being measured. Ideal voltmeters have infinite resistance so that no charge flows through them.
Nonideal ammeters and voltmeters will change the properties of the circuit being measured.
See also
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