Last updated on April 20th, 2023 at 04:54 pm
In this post, we will study a simple parallel circuit. Then we will derive the equivalent resistance formula in a parallel circuit. Using this formula, we will solve numerical problems related to parallel circuits.
Simple Parallel circuit | Resistance in Parallel | Resistors in parallel
A simple parallel circuit is shown below. Three resistances (resistors) are in parallel in this circuit. The supply voltage is applied in parallel to these parallel resistors.
The equivalent resistance in a parallel circuit is found using Kirchoff’s junction rule, which states that the sum of the currents entering a branch (or junction) must be equal to the sum of the currents leaving a branch (or junction).
This is illustrated in Figure 2 (the black dot represents the branch point). See how 10 A current is entering the junction and again 6+4 =10 A current is coming out of it through two branches. The concept of a circuit branch is necessary to understand parallel circuits.
Derivation of the Equivalent resistance in a parallel circuit
A simple parallel circuit would look like Figure 3. The circuit current I splits into two currents, I1 and I2 (as measured by ammeters A1 and A2). Those two currents must add up to the total circuit current I coming from the battery.
However, experiments show that the potential drops across each resistor in parallel are the same.
Thus, we have a parallel circuit:
VT = V1 = V2 = V3 = ….. (a)
I = I1 + I2 + I3 +… (b) [ current I gets distributed among parallel paths.]
Now, using Ohm’s Law in equation (b) we get,
VT/Req = V1/R1 + V2/R2 + V3/R3 ..
=> VT/Req = VT/R1 + VT/R2 + VT/R3 .. [ as we know VT = V1 = V2 = V3 ]
1/Req = 1/R1 + 1/R2 + 1/R3 .. (c)
Req = Equivalent resistance of the resistors in parallel.
When there are just 2 parallel paths then equation (c) looks like this:
1/Req = 1/R1 + 1/R2
=>Equivalent resistance in a parallel circuit Req = R1R2 /( R1 + R2 ) (d)
Equivalent resistance formula for a parallel circuit
Equivalent resistance in a parallel circuit of 2 resistances in parallel: Req = R1R2 /( R1 + R2 )
Numerical Problems on equivalent resistance of parallel circuits | Parallel circuit problems with Solutions
Two resistors with resistances of 5 Ω and 20 Ω each are connected in parallel to a 16-V battery. Calculate the equivalent resistance of the circuit, the total circuit current, and the current in each branch of the circuit.
Equivalent resistance = Req = (5×20)/(5+20) = 100/25 = 4 ohm
VT = 16 volts
Total circuit current = VT / Req = 16/4 amp = 4 amps
Current through 5 ohm branch = VT /5 amps = 16/5 amps = 3.2 amps
Current through 20 ohm branch = VT /20 amps = 16/20 amps = 4/5 amps=0.8 amps
Three resistors of 30 Ω, 15 Ω, and 10 Ω are connected in parallel to a 20-V battery. Calculate the equivalent resistance of the circuit, the total current for the circuit, and the current flowing through each resistor.
Let’s first find out the eq resistance of 30 ohms and 15 ohms = (30×15)/(30 +15) ohm= (30×15)/(45) ohm = 10 ohm
Then we will find out the final eq resistance by using the above-derived resistance 10 ohms and the 3rd resistance (another 10 ohms again) = (10×10)/(10 + 10) ohm =100/20 ohm = 5 ohm
so Req = 5 ohm
VT = 20 volts
Total circuit current = VT / Req =20/5 Amps = 4 A
Current through 30 ohm path = 20/30 A=2/3 A
Current through 15 ohm path = 20/15 A = 4/3 A
Current through 10 ohm path = 20/10 A = 2 A
Ohm’s Law Numerical problem worksheet – test your preparation