Resistance in Parallel

Connecting some electrical resistor between two common points having same potential difference for each resistor between two points is known parallel combination of resistors.

Parallel resistance is shown in figure. Here, R_1, R₂, R₃ resistors one point is connected at A point and another point is connected at B.

Equivalent resistance: Consider, V_A and V_B are the potential difference of A and B point and V_A>V_B. Total current I supplies. Entering current divides into 3 branches as I₁, I₂, I₃ and flow over R_1, R₂, R_3. Then I meet at B point.

Consider, I₁, I₂ and I₃ are the currents across R_1, R₂ and R₃ resistors.

I = I₁+ I₂+I₃ ————- (i)

Applying Ohm’s law between A and B points for three branches,

I₁ = V_A-V_B / R₁

I₂ = V_A-V_B / R₂

I₃ = V_A-V_B / R₃

Substituting values of  I₁, I₂ and I₃ in equation in (i),

I = V_A-V_B / R₁ + V_A-V_B / R₂ + V_A-V_B / R₃

I = (V_A-V_B) (1 / R₁ + 1/ R₂ +  1 / R₃ ) ————- (ii)

Replacing R_P for R_1, R₂ and R₃ where potential difference is same for A and B points. Total current I will be flowed across R_P resistance.

Applying Ohm’s law again for equivalent resistance parallel we get,

I = V_A-V_B / R_P ————- (iii)

From (ii) & (iii)

V_A-V_B / R_P = (V_A-V_B) (1 / R₁ + 1/ R₂ +  1 / R₃ )

Or,        1 / Rp = 1 / R₁ + 1/ R₂ +  1 / R₃ +…………….

This is the parallel equation of resistance.

For n number of resistor which are parallel connected

1 / Rp = 1 / R₁ + 1/ R₂ +  1 / R₃ +……………..+1 / R_n