AcademyElectric Potential
Academy
Electric Potential Energy
Level 1 - Physics topic page in Electric Potential.
Principle
Electric potential energy is stored energy associated with the positions of charges in a conservative electric interaction.
Notation
\(U\)
electric potential energy
\(\mathrm{J}\)
\(\Delta U\)
change in electric potential energy
\(\mathrm{J}\)
\(W_{\mathrm{elec}}\)
work done by the electric force
\(\mathrm{J}\)
\(q_1,q_2\)
point charges
\(\mathrm{C}\)
\(r\)
separation between point charges
\(\mathrm{m}\)
\(k\)
Coulomb constant
\(\mathrm{N\,m^{2}\,C^{-2}}\)
Method
Derivation 1: Link field work to stored energy
The electric force from fixed charges is conservative. When that force does positive work, the stored electric potential energy decreases.
Conservative-force rule
\[\Delta U=-W_{\mathrm{elec}}\]
Energy conservation
\[\Delta K+\Delta U=0\]
This form applies when the electric force is the only force doing work.
External quasistatic work
\[W_{\mathrm{ext}}=\Delta U\]
A slow external agent supplies the energy change without changing kinetic energy.
Derivation 2: Choose the zero of energy
For two point charges, take \(U=0\) when the charges are infinitely far apart. Bringing the charges to separation \(r\) gives the standard point-charge potential energy.
Zero reference
\[U(\infty)=0\]
Point-charge energy
\[U(r)=k\frac{q_1q_2}{r}\]
Like charges
\[q_1q_2>0\Rightarrow U>0\]
Work must be supplied to assemble like charges from infinity.
Opposite charges
\[q_1q_2<0\Rightarrow U<0\]
The electric force releases energy as opposite charges come together.
Rules
These are the compact energy relations.
Field work
\[\Delta U=-W_{\mathrm{elec}}\]
External work
\[W_{\mathrm{ext}}=\Delta U\quad\text{for quasistatic assembly}\]
Point-charge energy
\[U=k\frac{q_1q_2}{r}\quad(U(\infty)=0)\]
Electric-only energy
\[K+U=\mathrm{constant}\]
Examples
Question
Two charges
\[+2.0\,\mathrm{nC}\]
and \[+5.0\,\mathrm{nC}\]
are \[0.30\,\mathrm{m}\]
apart. Find their electric potential energy using \[U(\infty)=0\]
Answer
\[U=k\frac{q_1q_2}{r}=\frac{(8.99\times10^9)(2.0\times10^{-9})(5.0\times10^{-9})}{0.30}=3.0\times10^{-7}\,\mathrm{J}\]
The energy is positive because the charges are like-signed.Checks
- Electric potential energy is a property of a charge arrangement, not of one isolated charge alone.
- The sign of \(U=kq_1q_2/r\) comes from the charge product.
- Positive electric work means potential energy decreases.
- Always state the zero reference when quoting a potential energy.