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Resistors in Parallel 

When two resistors are connected in parallel, as shown to the right, the same voltage appears across each resistor. However, each resistor provides its own path for the flow of current. If the resistors have different resistance values, they will carry different amounts of current, each in accordance with Ohm's Law.
As a result, we can calculate the currents through each resistor, and the total current I, as:
I_{1} = E ÷ R_{1}
I_{2} = E ÷ R_{2}
I = I_{1} + I_{2}
Now let's apply Ohm's Law again, and solve the above equation for total resistance:
E  =_{ }  E  +_{ }  E 
R_{T}  R_{1}  R_{2} 
Since E is the same everywhere in the circuit, we can multiply both sides of the equation by 1/E and thus remove it. Then we solve for R_{T}, the total circuit resistance:
1  =_{ }  1  +_{ }  1 
R_{T}  R_{1}  R_{2}  
R_{T} =  1  
1  +  1  
R_{1}  R_{2}  
R_{T} =  1  
R_{2}  +  R_{1}  
R_{1}R_{2}  R_{2}R_{1}  
R_{T} =  1  
R_{1} + R_{2}  
R_{1} × R_{2}  
R_{T} =  R_{1} × R_{2}  
R_{1} + R_{2} 


 
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