Chapter 28: Direct Current Circuits

Chapter 28: introduces 'Direct Current Circuits' also known as DC circuits and "Resistor Capacitor Circuits, also known as RC circuits. 

DEFINITIONS:

Resistance is a function of temperature, it is the amount of voltage per unit of current across a resistor
UNIT: ohms

the emf of a battery is called the electromotive force 

emf of a battery is equal to the voltage difference across its terminals, when the current (I) = 0. This voltage situation is described as "open circuit voltage".

Equivalent resistance: when a set of resistors are connected and represented as one in a circuit diagram

Resistors in parallel:

• 1/Req = 1/R1 + 1/R2 + 1/R3 + ....

Resistors in series:

• Req = R1 + R2 +....

Circuits involving more than one loop are analyzed with Kirchoff's Rules: 

1. Junction Rule: at any junction the sum of the currents (I) must equal zero:

• Sum I = 0

2. Loop Rule: the sum of the potential differences across all elements around any circut loop must be zero:

• Sum (Vf-Vi) = 0

Steps to solve: 

1.  determine the number of loops 

loop 1: abefa

loop 2: bcdeb

       2.   "travel" each loop and write the Sum of Delta V = 0 for EACH CIRCUIT ELEMENT

Delta V = Vdownstream - Vupstream

for an ideal battery: I in the -  -->  + emf direction

• Delta V = + emf 

for an ideal battery: I in the +  -->  - emf direction

• Delta V = - emf 

for a resistor: 

I in the + ---> - direction 

• Delta VR = -IR

           3. set each loop equation equal to zero. solve for I. plug into the Junction Rule (1). Multiply the                  remaining equation by any common factor. Solve for a real value for I. Plug back in to find the                other value for I.

If a capacitor is CHARGED by a battery through a resistor (R), the charge on the capacitor and the current in the circuit vary in time:

• q(t) = Q(1-e^(-t/RC)) 

• I(t) = [emf / R]e^(-t/RC)

Q = C(emf) = maximum charge on the capacitor 

RC = time constant, T of the circuit 

****this is a differential equation. Why?

Answer: there is a derivative in the solution

If a charged capacitor is DISCHARGED through a resistor with resistance, R, the charge and current decrease exponentially in time: 

• q(t) = Qe^(-t/RC)

• I(t) = -Iie^(-t/RC)

Q = initial charge on the capacitor 

Ii = Q/RC = initial current in the circuit