A four-ampere current passes through a 10-ohm resistor. How much is the power of the resistor?
Okay, so we know that we’ve got a resistor with a resistance of 10 ohms, which we’ve labeled as 𝑅. We’re also told that there’s a current of four amperes through the resistor, and we’ve labeled this current as 𝐼. In order for there to be a current through it, then this resistor must be part of a complete circuit. So, for example, we could imagine the simplest possible case, which is the resistor connected in series with a cell.
We’re being asked to work out the power of the resistor. And we can recall that the electrical power 𝑃 of a component or the power that it dissipates or transfers to its surrounding environment is equal to the current 𝐼 through that component multiplied by the potential difference 𝑉 across it. In this case, we already know the value of the current 𝐼 through the resistor.
The potential difference 𝑉 is the value that will be measured by a voltmeter connected in parallel across the resistor like this. But we don’t know what this value is. However, we do know the resistance of the resistor. And we can recall that Ohm’s law links a component’s resistance, the current through it, and the potential difference across it. Specifically, Ohm’s law tells us the potential difference 𝑉 is equal to current 𝐼 multiplied by resistance 𝑅.
We can then use this Ohm’s law equation to replace the 𝑉 in the power equation by 𝐼 times 𝑅. So we’d be replacing the quantity 𝑉 that we don’t know the value of by two quantities 𝐼 and 𝑅 that we do know values for. If we take our equation for the power 𝑃 and we use Ohm’s law to replace the quantity 𝑉 with 𝐼 times 𝑅, then we have that the power 𝑃 is equal to 𝐼 times 𝐼 times 𝑅, which we can write more simply as 𝑃 is equal to 𝐼 squared times 𝑅.
This equation tells us that the power 𝑃 of the resistor is equal to the square of the current 𝐼 through the resistor multiplied by its resistance 𝑅. We can now go ahead and substitute our values for 𝐼 and 𝑅 into this equation to calculate the value of the power 𝑃. When we do this, we find that 𝑃 is equal to the square of four amperes, that’s the current 𝐼, multiplied by 10 ohms, the resistance 𝑅.
Our current with units of amperes and a resistance in units of ohms will mean that we calculate a power in units of watts. Evaluating the expression, the square of four is 16, so we’ve got 16 multiplied by 10 watts, which works out as a power of 160 watts. So our answer to this question is that the power of the resistor is 160 watts.