# Direct Current Circuits

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```					                               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.

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