To find the proportion of the current flowing through the 12-ohm resistor in a parallel circuit with a 12-ohm and a 4-ohm resistor connected across a 12V supply, we can use Ohm's Law and the concept of parallel resistances.
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Calculate the total current (I_total) supplied by the voltage source: The equivalent resistance (R_eq) for resistors in parallel is given by the formula:
\[ \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} \]
where \( R_1 = 12 , \Omega \) and \( R_2 = 4 , \Omega \).
\[ \frac{1}{R_{eq}} = \frac{1}{12} + \frac{1}{4} \]
Convert \( \frac{1}{4} \) to have a common denominator:
\[ \frac{1}{R_{eq}} = \frac{1}{12} + \frac{3}{12} = \frac{4}{12} = \frac{1}{3} \]
Therefore:
\[ R_{eq} = 3 , \Omega \]
Now, using Ohm's Law:
\[ I_{total} = \frac{V}{R_{eq}} = \frac{12V}{3\Omega} = 4A \]
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Calculate the current through each resistor: Now we can find the current through each resistor using Ohm's Law again.
For the 12-ohm resistor:
\[ I_{12\Omega} = \frac{V}{R_1} = \frac{12V}{12\Omega} = 1A \]
For the 4-ohm resistor:
\[ I_{4\Omega} = \frac{V}{R_2} = \frac{12V}{4\Omega} = 3A \]
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Proportion of the current through the 12-ohm resistor: The proportion of the total current that flows through the 12-ohm resistor is given by:
\[ \text{Proportion} = \frac{I_{12\Omega}}{I_{total}} = \frac{1A}{4A} = \frac{1}{4} \]
Thus, the proportion of the current that flows through the 12-ohm resistor is \( \frac{1}{4} \) or 25%.