On a particular day, it takes 3.0 times 10 to the third joules of electrical energy to start a truck’s engine. What is the capacitance of a capacitor that could store that amount of energy at 8.0 volts?
In this scenario, we have three quantities to keep in mind. One is an amount of energy. One is a potential difference. And one, the one we want to solve for, is capacitance. So potential difference, energy, and capacitance.
It turns out there is a mathematical relationship that connects all three of these quantities. That relationship says that the energy stored in a capacitor, we can call it 𝑢, is equal to one-half the capacitance of the capacitor 𝑐 multiplied by the potential difference across it 𝑣 squared.
Now in our case, we’re told the energy as well as the potential difference. And we want to solve for the capacitance 𝑐. So that we’ll take a bit of rearranging. When we do that, we find that the capacitance is equal to two times the energy 𝑢 divided by 𝑣 squared.
From the information given to us in our problem statement, we see that 𝑢 is equal to 3.0 times 10 to the third joules. And 𝑣, our potential difference, is 8.0 volts. So we’re ready to plug these values in and solve for capacitance 𝑐.
When we do and enter this expression on our calculator, to two significant figures, we find a result of 94 farads. That’s the capacitance that a capacitor would need to store this amount of energy at 8.0 volts. And it’s a huge capacitance.