Direct Current Circuits A direct current (DC) circuit is defined as a type of circuit in which charge flows smoothly, connected to a potential source, with simple circuit elements connected in series or parallel. EMF A device which increases the potential energy of charge circulating within an electric circuit is termed a source of emf, symbolized as and sometimes referred to as “electromotive force”. As an example, a battery is a source of emf, converting chemical potential energy into electrical potential energy. The potential across the terminals of a battery is not in general equal to the battery emf, due to the non-zero internal resistance within a battery. Terminal voltage for a battery is given as: V I r Where is the emf of the battery, I is the current, and r is the internal resistance of the battery. In most situations, r is small enough to be ignored in most applications. Combinations of Resistors Circuit elements which resist current flow, and allow work to be done using electric current, may be combined in series or parallel. Series resistors: Each resistor follows the last, with one just one loop through the source of emf. The equivalent resistance of two or more resistors combined in series is equal to the sum of the resistances: Req R1 R2 R3 R N The current, I, is the same through each resistor, a consequence of conservation of charge. The potential differences across a series resistor will be different, except the special case of resistors which are equal. Potentials add up, equaling the total potential across the combination: V = V1 + V2 + +VN. Parallel resistors: Each resistor is connected to the source of emf. The equivalent resistance of a set of parallel resistors is expressed: 1 1 1 1 Req R1 R2 RN The potential difference across each resistor is the same: V1 = V2 = V3 = … = VN, because each resistor is connected to the source of emf. The current passing through each resistor will be different, and will add to the total current: I = I1 + I2 + I3 + +IN, where I1 = V / R1 , etc Complex DC Circuits: Circuits may contain large numbers of resistors, in which the equivalent resistance cannot be determined. These circuits may be analyzed using Kirchhoff’s rules. Junction rule: The sum of the currents entering a connecting point in a circuit = the sum of the currents leaving the connection. This rule is a consequence of conservation of charge – charge flowing into one end of must flow out the other. Loop rule: The sum of the potential differences across all elements in a closed loop must be zero. This rule is a consequence of conservation of energy. For a given complex circuit, with unknown currents, resistances, and potentials, the circuit can be solved by using Kirchhoff’s rules to construct a set of independent equations, equaling the number of unknowns, and solving this equation set. RC Circuits Circuits with a source of emf, resistors and a capacitor behave in a time-dependent manner. Time dependent means that the current and charge are not “steady-state”, but rather change with time. Once the circuit is closed, the capacitor will take a finite amount of time to charge up and discharge. Charge on a capacitor increases according to: q Q(1 e t / RC ) , where Q is the final total charge. The product RC is known as the time constant (symbolized as ) of the circuit, and is equal to the time required to charge the capacitor to (1/e) 1 of its final value ( = 63.2%). An RC circuit will also discharge in a time-dependent manner according to: q Qe t / RC where Q is the total charge in the system, and q is the charge at time t.