Write the expression for current in a parallel combination.
2 Answers
In a parallel circuit, the total current is the sum of the currents through each parallel branch. If you have a parallel combination of \( n \) resistors with resistances \( R_1, R_2, \ldots, R_n \) connected to a voltage source \( V \), the current through each resistor \( i_k \) is given by Ohm's Law:
\[ i_k = \frac{V}{R_k} \]
where \( k \) ranges from 1 to \( n \).
The total current \( I \) in the parallel circuit is:
\[ I = i_1 + i_2 + \cdots + i_n = \frac{V}{R_1} + \frac{V}{R_2} + \cdots + \frac{V}{R_n} \]
If you want to express the total current in terms of the total equivalent resistance \( R_{eq} \) of the parallel combination, you first calculate \( R_{eq} \) as:
\[ \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots + \frac{1}{R_n} \]
Then, the total current \( I \) is:
\[ I = \frac{V}{R_{eq}} \]
\[ i_k = \frac{V}{R_k} \]
where \( k \) ranges from 1 to \( n \).
The total current \( I \) in the parallel circuit is:
\[ I = i_1 + i_2 + \cdots + i_n = \frac{V}{R_1} + \frac{V}{R_2} + \cdots + \frac{V}{R_n} \]
If you want to express the total current in terms of the total equivalent resistance \( R_{eq} \) of the parallel combination, you first calculate \( R_{eq} \) as:
\[ \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots + \frac{1}{R_n} \]
Then, the total current \( I \) is:
\[ I = \frac{V}{R_{eq}} \]
When dealing with electrical circuits, understanding the behavior of current in parallel combinations is crucial. In a parallel circuit, multiple components are connected across the same two points, so each component experiences the same voltage across it.
### Expression for Current in a Parallel Combination
Consider a parallel circuit with \( n \) resistors connected across the same voltage source. The total current flowing from the voltage source is the sum of the currents through each resistor. Here's how you can derive the expression for the total current:
1. **Identify the Voltage Across Each Resistor:**
In a parallel circuit, the voltage across each resistor is the same and equal to the voltage of the source \( V \).
2. **Calculate the Current Through Each Resistor:**
Using Ohm's Law (\( I = \frac{V}{R} \)), the current through a resistor \( R_i \) is:
\[
I_i = \frac{V}{R_i}
\]
where \( I_i \) is the current through the \( i \)-th resistor and \( R_i \) is its resistance.
3. **Sum the Currents Through All Resistors:**
The total current \( I_{total} \) supplied by the source is the sum of the currents through all resistors:
\[
I_{total} = \sum_{i=1}^{n} I_i
\]
Substituting \( I_i = \frac{V}{R_i} \) into this formula, we get:
\[
I_{total} = \sum_{i=1}^{n} \frac{V}{R_i}
\]
4. **Factor Out the Voltage:**
Since \( V \) is common for all resistors, it can be factored out:
\[
I_{total} = V \left(\sum_{i=1}^{n} \frac{1}{R_i}\right)
\]
### Example with Two Resistors
If there are two resistors \( R_1 \) and \( R_2 \) in parallel, the total current \( I_{total} \) can be computed as follows:
1. Calculate the current through each resistor:
\[
I_1 = \frac{V}{R_1}
\]
\[
I_2 = \frac{V}{R_2}
\]
2. Sum the currents:
\[
I_{total} = I_1 + I_2 = \frac{V}{R_1} + \frac{V}{R_2}
\]
3. Factor out the voltage:
\[
I_{total} = V \left(\frac{1}{R_1} + \frac{1}{R_2}\right)
\]
In summary, the total current \( I_{total} \) in a parallel combination of resistors is given by:
\[
I_{total} = V \left(\sum_{i=1}^{n} \frac{1}{R_i}\right)
\]
where \( V \) is the voltage across the resistors and \( R_i \) are the resistances of the individual resistors.
### Expression for Current in a Parallel Combination
Consider a parallel circuit with \( n \) resistors connected across the same voltage source. The total current flowing from the voltage source is the sum of the currents through each resistor. Here's how you can derive the expression for the total current:
1. **Identify the Voltage Across Each Resistor:**
In a parallel circuit, the voltage across each resistor is the same and equal to the voltage of the source \( V \).
2. **Calculate the Current Through Each Resistor:**
Using Ohm's Law (\( I = \frac{V}{R} \)), the current through a resistor \( R_i \) is:
\[
I_i = \frac{V}{R_i}
\]
where \( I_i \) is the current through the \( i \)-th resistor and \( R_i \) is its resistance.
3. **Sum the Currents Through All Resistors:**
The total current \( I_{total} \) supplied by the source is the sum of the currents through all resistors:
\[
I_{total} = \sum_{i=1}^{n} I_i
\]
Substituting \( I_i = \frac{V}{R_i} \) into this formula, we get:
\[
I_{total} = \sum_{i=1}^{n} \frac{V}{R_i}
\]
4. **Factor Out the Voltage:**
Since \( V \) is common for all resistors, it can be factored out:
\[
I_{total} = V \left(\sum_{i=1}^{n} \frac{1}{R_i}\right)
\]
### Example with Two Resistors
If there are two resistors \( R_1 \) and \( R_2 \) in parallel, the total current \( I_{total} \) can be computed as follows:
1. Calculate the current through each resistor:
\[
I_1 = \frac{V}{R_1}
\]
\[
I_2 = \frac{V}{R_2}
\]
2. Sum the currents:
\[
I_{total} = I_1 + I_2 = \frac{V}{R_1} + \frac{V}{R_2}
\]
3. Factor out the voltage:
\[
I_{total} = V \left(\frac{1}{R_1} + \frac{1}{R_2}\right)
\]
In summary, the total current \( I_{total} \) in a parallel combination of resistors is given by:
\[
I_{total} = V \left(\sum_{i=1}^{n} \frac{1}{R_i}\right)
\]
where \( V \) is the voltage across the resistors and \( R_i \) are the resistances of the individual resistors.