Question

The highest magnetic fields in the world are generated when large arrays, or "banks," of capacitors are discharged through the copper coils of an electromagnet. At the National High Magnetic Field Laboratory, the total capacitance of the capacitor bank is 32 mF. These capacitors can be charged to 16 kV.What is the energy stored in the capacitor bank when it is fully charged?When discharged, the entire energy from this bank flows through the magnet coil in 10 ms. What is the average power delivered to the coils during this time?

Answer 1

1. $4.1\cdot 10^7 J$

The energy stored in a capacitor is given by

$U=\frac{1}{2}CV^2$

where

C is the capacitance

V is the potential difference across the capacitor

In this problem, we have

$C=32 mF = 0.032 F$ is the capacitance of the capacitor

$V=16 kV=16000 V$ is the p.d. across the capacitor

Solving the equation, we find the energy stored:

$U=\frac{1}{2}(0.032 F)(16000 V)^2=4.1 \cdot 10^7 J$

2. $4.1\cdot 10^9 W$

The average power delivered to the coils is equal to the ratio between the energy released and the time taken:

$P=\frac{E}{t}$

where

$E=4.1 \cdot 10^7 J$ is the energy released by the capacitor

$t = 10 ms = 0.01 s$ is the time taken

Substituting into the formula, we find

$P=\frac{4.1\cdot 10^7 J}{0.01 s}=4.1\cdot 10^9 W$