Electric Potential Energy
Introduced| English |
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| electric potential energy/ɪˈlektrɪk pəˈtenʃl ˈenədʒi/ |
Pushing two like charges together is like squeezing a spring
- Push two positive charges toward each other and they resist harder and harder.
- You are doing work against their repulsion, and that work is stored.
- The stored energy is electric potential energy 电势能 — released the moment you let go.
- It is the electric cousin of a compressed spring or a lifted weight.
The stored-energy formula
- For two point charges, the electric potential energy is $U = \dfrac{k q_1 q_2}{r}$.
- Note it falls off as $\dfrac{1}{r}$ — one power of distance, unlike the $\dfrac{1}{r^2}$ force.
- For like charges $U$ is positive (you did work to push them together).
- For opposite charges $U$ is negative (they pulled together on their own).
Energy in the field
See the field a charge builds; moving another charge through it stores or releases energy.
Two $+1\ \text{C}$ charges are $2\ \text{m}$ apart ($k = 9\times10^9$). What is their potential energy, in $\text{J}$? (Give the coefficient of $10^9$.)
$U = kq_1q_2/r = 9\times10^9 / 2 = 4.5\times10^9\ \text{J}$.
Electric potential energy between two point charges depends on distance as:
$U = kq_1q_2/r$ falls as $1/r$ — one power, unlike the $1/r^2$ force.
For two like charges, the electric potential energy is:
You must do positive work to push like charges together, so $U > 0$.
For two point charges, halving their separation doubles their electric potential energy.
$U \propto 1/r$, so halving $r$ doubles $U$.
Work and energy
- The work you do moving a charge equals the change in its potential energy.
- Move a charge to where the field pushes it, and its PE drops (energy released).
- Move it against the field, and its PE rises (energy stored).
- Only changes in potential energy matter, so you may set any convenient zero.
The work done moving a charge equals the ____ in its potential energy.
Work $= \Delta U$ — only changes in PE matter.
Just like gravity
- Electric PE mirrors gravitational PE: both store energy in a configuration.
- Lifting a mass raises gravitational PE; separating opposite charges raises electric PE.
- Release either and the stored energy becomes kinetic energy.
- The same energy-conservation tools apply to both.
Select all correct statements about electric potential energy.
Electric PE mirrors gravitational PE, goes as $1/r$, and is positive for like charges. The force (not PE) goes as $1/r^2$.
Electric potential energy goes as $\dfrac{1}{r}$, while the force goes as $\dfrac{1}{r^2}$ — don't confuse the two powers. And watch the signs: like charges have positive $U$, opposite charges negative $U$.
Two $+1\ \text{C}$ charges are $2\ \text{m}$ apart, with $k$ taken as $9\times10^9$.
- $U = \dfrac{kq_1q_2}{r} = \dfrac{9\times10^9 \times 1 \times 1}{2} = 4.5\times10^9\ \text{J}$.
Bring them to $1\ \text{m}$ apart and $U$ doubles to $9\times10^9\ \text{J}$ — you must do that much extra work.
Electric potential energy is the energy stored in an arrangement of charges: $U = \dfrac{kq_1q_2}{r}$ (note $1/r$, not $1/r^2$). Like charges give positive $U$, opposites negative. Moving a charge does work equal to the change in $U$, just like gravitational PE.