Electrical Power and Resistors in Circuits
Electrical Power in Resistors
Power
Power is the rate at which energy is transferred or converted.
In electrical circuits, power is dissipated aspower is dissipated as thermal energy in resistors.
The formula for power in a resistor is:$$
P = IV
$$ where:
- P is the power (in watts, W)
- I is the current (in amperes, A)
- V is the potential difference (in volts, V)
Using Ohm’s Law ($V = IR$), we can express power in two other forms:
- In terms of current and resistance:$$
P = I^2R
$$ - In terms of voltage and resistance:$$
P = \frac{V^2}{R}
$$
Choose the power formula that matches the information you have.
- If you know the current and resistance, use $P = I^2R$.
- If you know the voltage and resistance, use $P = \frac{V^2}{R}$.
A resistor has a resistance of 5 Ω and a current of 2 A flowing through it. Calculate the power dissipated.
Solution
Using $P = I^2R$:
$$
P = (2 \, \mathrm{A})^2 \times 5 \, \Omega = 20 \, \mathrm{W}
$$
Resistors in Series Circuits
Series circuit
In a series circuit, resistors are connected end-to-end, forming a single path for current.
Key Characteristics of Series Circuits
- Current: The current is the same through all resistors. $$
I = I_1 = I_2 = I_3 = \ldots
$$ - Voltage: The total voltage across the circuit is the sum of the voltages across each resistor. $$
V = V_1 + V_2 + V_3 + \ldots
$$ - Resistance: The total resistance is the sum of the individual resistances. $$
R_s = R_1 + R_2 + R_3 + \ldots
$$
In series circuits, adding more resistors increases the total resistance, which decreases the current for a given voltage.
Three resistors with resistances 2 Ω, 4 Ω, and 6 Ω are connected in series. Calculate the total resistance and the current if the total voltage is 12 V.
Solution
- Total Resistance:$$
R_s = 2 \, \Omega + 4 \, \Omega + 6 \, \Omega = 12 \, \Omega
$$ - Current (using Ohm’s Law, $I = \frac{V}{R}$):$$
I = \frac{12 \, \mathrm{V}}{12 \, \Omega} = 1 \, \mathrm{A}
$$
Resistors in Parallel Circuits
Parallel circuit
In a parallel circuit, resistors are connected across the same two points, providing multiple paths for current.
Key Characteristics of Parallel Circuits
- Voltage: The voltage across each resistor is the same. $$
V = V_1 = V_2 = V_3 = \ldots
$$ - Current: The total current is the sum of the currents through each resistor. $$
I = I_1 + I_2 + I_3 + \ldots
$$ - Resistance: The reciprocal of the total resistance is the sum of the reciprocals of the individual resistances. $$
\frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \ldots
$$
In parallel circuits, adding more resistors decreases the total resistance, which increases the current for a given voltage.
Three resistors with resistances 2 Ω, 4 Ω, and 6 Ω are connected in parallel. Calculate the total resistance and the current if the total voltage is 12 V.
Solution
- Total Resistance:$$
\frac{1}{R_p} = \frac{1}{2} + \frac{1}{4} + \frac{1}{6} = \frac{6}{12} + \frac{3}{12} + \frac{2}{12} = \frac{11}{12}
$$ $$R_p = \frac{12}{11} \, \Omega \approx 1.09 \, \Omega
$$ - Total Current (using Ohm’s Law, $I = \frac{V}{R}$):$$
I = \frac{12 \, \mathrm{V}}{1.09 \, \Omega} \approx 11.01 \, \mathrm{A}
$$
Combining Series and Parallel Circuits
- Many circuits contain a combination of series and parallel resistors' combination of series and parallel resistors.
- To analyze these circuits:
- Simplify the circuit by reducing series and parallel groups step-by-step.
- Calculate the total resistance.
- Use Ohm’s Law to find the total current.
- Work backward to find the current, voltage, or power for individual resistors.
- A common mistake is to assume that current is the same in parallel resistors.
- Remember, current splits in parallel paths, but voltage remains constant across each path.
Consider a circuit with three resistors: 2 Ω and 4 Ω in series, and a 6 Ω resistor in parallel with the series combination. The total voltage is 12 V. Calculate the current through each series resistor.
Solution
- Simplify the Series Resistors:$$
R_s = 2\ \Omega + 4\ \Omega = 6\ \Omega
$$ - Combine with the Parallel Resistor:$$
\frac{1}{R_{\mathrm{total}}} = \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3}
$$$$
R_{\mathrm{total}} = 3\ \Omega
$$ - Calculate the Total Current: $$
I = \frac{12\ \mathrm{V}}{3\ \Omega} = 4\ \mathrm{A}
$$ - Find the Voltage Across the Series Resistors:$$
V_s = IR_s = 4\ \mathrm{A} \times 6\ \Omega = 24\ \mathrm{V}
$$ - Find the Current Through Each Series Resistor (same current in series):
- 2 Ω resistor: $I = 4\ \mathrm{A}$
- 4 Ω resistor: $I = 4\ \mathrm{A}$
