Parallel Circuits
=================



  .. tikz:: Circuit with two resistors in parallel
     :xscale: 35
	      
     \draw (0,0)
     to[voltage source, v=$U$] (0,3)
     to[short, i=$I$] (2,3)
     to[R=$R_1$, i=$I_1$] (2,0)
     to[short] (0,0);
     \draw (2,3)
     to[short] (4,3)
     to[R=$R_2$, i=$I_2$] (4,0)
     to[short] (2,0);
     \draw (5,0) to (5,0);


Consider a circuit with two resistors in parallel. It is a closed
circuit and current will flow. The voltage drop over both resistors is
the same and equal to the battery voltage $U$.  Using Ohm's law we
then can calculate the current flowing through the resistors:

.. math::
   I_1 = \frac{U}{R_1}\\
   I_2 = \frac{U}{R_2}

The total current flowing from the battery through the circuit is the
sum of $I_1$ and $I_2$:

.. math::
   I = I_1 + I_2 = \frac{U}{R_1} + \frac{U}{R_2}

i.e. the two resistors in parallel act as one resistor with resistance
$R$ such that

.. math::
   \frac{1}{R} =  \frac{1}{R_1} + \frac{1}{R_2}

Our analysis above can be done for an arbitrary number of resistors in
a parallel circuit.

*In a parallel connection of resistors the voltage across all
resistors is the same, the current flowing through each resistor is
dependent on its resistance in relation to all other resistances.*