Use Kirchhoff's junction theorem to explain why the total equivalent resistance of a circuit is reduced, not increased, by connecting a second resistor in parallel to another resistor.
Kirchhoff's junction theorem, also known as Kirchhoff's current law, states that at any junction point in a circuit, the sum of incoming currents is equal to the sum of outgoing currents. To explain why the total equivalent resistance of a circuit is reduced by connecting a second resistor in parallel, we can use this theorem.
When resistors are connected in parallel, they share the same voltage across them. This means that the potential difference across each resistor is the same.
Let's consider two resistors, R1 and R2, connected in parallel. The total current that flows into this junction is equal to the sum of the currents passing through each resistor:
I_total = I1 + I2
According to Ohm's law, the current flowing through a resistor can be calculated using the formula:
I = V / R
Where I is the current, V is the voltage, and R is the resistance.
Since both R1 and R2 have the same voltage across them, we can rewrite the previous equation as:
I_total = V / R1 + V / R2
We can factor out V:
I_total = V * (1 / R1 + 1 / R2)
Now, let's calculate the equivalent resistance of the combination of R1 and R2, denoted as Req. The equivalent resistance is defined as the single resistor that would replace the combination of R1 and R2, and it would produce the same total current in the circuit.
By rearranging the expression for I_total, we can solve for V:
I_total = V * (1 / R1 + 1 / R2)
V = I_total / (1 / R1 + 1 / R2)
By substituting this value of V back into Ohm's law, we can solve for Req:
I_total = V / Req
I_total / (1 / R1 + 1 / R2) = V / Req
Now we can rearrange this equation to solve for Req:
1 / Req = (1 / R1) + (1 / R2)
Finally, flip both sides of the equation:
Req = 1 / (1 / R1 + 1 / R2)
Notice that the equivalent resistance, Req, is less than both individual resistances, R1 and R2. Therefore, by connecting a second resistor in parallel to another resistor, the total equivalent resistance of the circuit is reduced.