Two resistors, each of 20 ohms, are connected in parallel and this combination is connected across a 40 V supply voltage.
10 ohm
Understanding Resistors in Parallel Circuits
This problem asks us to find the total resistance of a circuit containing two resistors connected in parallel. When components like resistors are connected in parallel, there are multiple paths for the electric current to flow.
Calculating Equivalent Resistance in Parallel
For resistors connected in parallel, the reciprocal of the total equivalent resistance (\(R_{eq}\)) is equal to the sum of the reciprocals of the individual resistances. The formula for two resistors \(R_1\) and \(R_2\) in parallel is:
\[ \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} \]
A more convenient formula for exactly two resistors in parallel can be derived from this:
\[ R_{eq} = \frac{R_1 \times R_2}{R_1 + R_2} \]
Applying the Formula to the Given Circuit
We are given two resistors, each with a resistance of 20 ohms. Let's denote them as \(R_1\) and \(R_2\).
- \(R_1 = 20 \text{ ohms}\)
- \(R_2 = 20 \text{ ohms}\)
Now, we can use the formula for two resistors in parallel to calculate the total resistance:
\[ R_{eq} = \frac{R_1 \times R_2}{R_1 + R_2} \]
Substitute the given values:
\[ R_{eq} = \frac{20 \, \Omega \times 20 \, \Omega}{20 \, \Omega + 20 \, \Omega} \]
Calculate the numerator and the denominator:
\[ R_{eq} = \frac{400 \, \Omega^2}{40 \, \Omega} \]
Perform the division:
\[ R_{eq} = 10 \, \Omega \]
So, the total resistance of the two 20-ohm resistors connected in parallel is 10 ohms.
The supply voltage (40 V) was provided but is not needed to calculate the total resistance of the circuit. It would be needed if we were asked to calculate the total current drawn from the supply using Ohm's Law (\(V = I \times R\)).
Summary of Calculation for Parallel Resistors
| Component | Resistance (\(\Omega\)) | Connection |
|---|---|---|
| Resistor 1 | 20 | Parallel |
| Resistor 2 | 20 | Parallel |
| Total Equivalent Resistance | 10 | - |
Revision Table: Resistor Connections
| Connection Type | Total Resistance (\(R_{total}\)) Formula | Characteristic |
|---|---|---|
| Series | \(R_{total} = R_1 + R_2 + ... + R_n\) | Resistors are connected end-to-end, current is the same through each. Total resistance is greater than the largest individual resistance. |
| Parallel | \(\frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + ... + \frac{1}{R_n}\) | Resistors are connected across the same two points, voltage is the same across each. Total resistance is less than the smallest individual resistance. |
Additional Information on Circuit Resistance
Understanding how to calculate the total resistance in a circuit is fundamental in electronics and physics. It helps predict the total current flow for a given voltage supply (using Ohm's Law) and understand how different components behave when connected together.
- Ohm's Law: Relates voltage (\(V\)), current (\(I\)), and resistance (\(R\)) with the formula \(V = I \times R\).
- Parallel Connection: Provides multiple paths for current. This is common in household wiring so that if one appliance is turned off, the others continue to receive power. Adding more resistors in parallel actually decreases the total resistance of the circuit.
- Identical Resistors in Parallel: If 'n' identical resistors, each with resistance 'R', are connected in parallel, their equivalent resistance is simply \(R_{eq} = R/n\). In our problem, two 20-ohm resistors in parallel give \(R_{eq} = 20 \, \Omega / 2 = 10 \, \Omega\).
Calculating the total resistance in a circuit is the first step in analyzing its behavior, including current distribution and power dissipation.
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