Lecture 6 Direct Current Circuits Chapter 18 Outline Energy Source in Circuits Resistor Combinations Kirchhoff’s Rules RC Circuits

Energy in Circuits Circuits are usually powered by batteries or generators, which are called sources of emf. The emf ( ℇ )stands for electromotive force and is not a fundamental force. ℇ is the work per unit charge and is measured in volts (V). ℇ equals to the potential difference across the battery if the battery’s internal resistance is neglected. Direct current is a current flowing in one direction through a circuit.

Emf and Circuit Parameters  V = ℇ  IR int  V = IR load ℇ = IR load + IR int ℇ I =  R load + R int

Emf ℇ is equal to the terminal voltage in the absence of the current. ℇ I = I 2 R load +I 2 R int The total power output of the source of emf is converted to: the power delivered to the load resistance + the power delivered to the internal resistance In all the problems, we assume that the internal resistance of the battery is negligible.

Resistors in Series The current (I) is the same in both resistors.  V 1 = IR 1,  V 2 =IR 2  V = IR 1 + IR 2 = I(R 1 +R 2 )  V = IR eq R eq = R 1 + R 2 The equivalent resistance of a series combination of resistors equals to the algebraic sum of the individual resistances. Example

Resistors in Parallel The potential differences across both resistors are the same (  V), as each is directly connected to the battery.both resistors Due to conservation of charge, I tot = I 1 + I 2  V  V  V I tot =  = I 1 + I 2 =  +    =  +  R eq R 1 R 2 R eq R 1 R 2

Kirchhoff’s Rules Circuit elements can be connected in various ways. For analysis of complex circuits, 2 rules proposed by Gustav Kirchhoff in 1845 can be applied.Kirchhoff 1.The sum of all the currents entering any junction point is equal to the sum of all the currents leaving that junction point (junction rule). 2.The sum of the potential differences across all the elements around any closed-circuit loop must be zero (loop rule).

Kirchhoff’s rules The junction rule is a consequence of conservation of charge, the loop rule is a consequence of conservation of energy. Application of Kirchhoff’s rules

RC Circuits Circuits which contain capacitors have time- dependent currents. Q = C ℇ Q and ℇ are the maximum charge and voltage across the capacitor C is the capacitance q(t) = Q(1  e  t/RC ) e =  = RC is the capacitor’s time constant

RC Circuits Another view of a RC circuit Problem: An uncharged capacitor and a resistor are connected in a series to an emf source. ℇ = 10 V, C = 20  F, R = 100 . Find: the circuit’s time constant, the maximum charge on the capacitor, the charge on the capacitor after 1 time constant. Solution:  = RC = ( 10 2  ) (2 10  5 F) = 2 10  3 c Q = C ℇ = (2 10  5 F) (10 V) = 2 10  4 C = 200  C q = Q = (200  C) = 126  C

Summary Direct current is current that flow in a circuit in one direction Kirchhoff’s rules are a set of instructions to analyze complex circuits. The rules are consequences of conservation of charge and energy. Circuits containing capacitors have time- dependent currents. The charge on the capacitor and time dependence of the current through it can be determined using the capacitor time constant.