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Electric Fields overview

Topic 2 of 7

Electric potential and energy

Electric potential is energy per charge at a point. Electric potential energy belongs to the interaction of charges, and its change depends on the sign of the charge being moved.

Choose the zero reference

Electric potential at a point is the work done per unit small positive test charge by an external force in bringing it from infinity to that point, without changing its kinetic energy. Set potential to zero at infinity for the isolated point-source examples here.

V = Q/(4πε0r) = KQ/r
1 V = 1 J/C

Potential is a scalar. It is positive for a positive point source and negative for a negative source with this reference. For a positive test charge, moving slowly towards a positive source requires positive external work against repulsion. Moving it towards a negative source permits negative external work while the field does positive work.

A few nanocoulombs at a few tenths of a metre, with K of order 1010 N m2/C2, suggests potential of order 102 V. The sign is a separate decision from that rough magnitude.

Multiply by the actual signed charge

For two point charges Q and q, the pair's electric potential energy, with zero at infinite separation, is:

UE = Qq/(4πε0r) = KQq/r = qV

UE is in joules, whereas V is in joules per coulomb. Unlike charges have negative interaction energy relative to infinite separation: separating them without increasing kinetic energy requires an energy input. A negative energy is meaningful with the stated reference.

Potential difference is the change in potential between two points. For unchanged q:

ΔV = Vfinal - Vinitial
ΔUE = qΔV

Worked signed energy change

Move a negative charge outwards

Keep source Q = +4.00 nC fixed. Move q = -2.00 nC slowly from 0.300 to 0.600 m, using K = 9.00 × 109 N m2/C2.

Alternative states of the same source and negative test charge
r / mSource V / VPair UE / J
0.300+120-2.40 × 10-7
0.600+60.0-1.20 × 10-7

A negative charge can gain energy as potential falls

The two q markers are alternative positions of the same -2.00 nC charge. The source Q = +4.00 nC is held fixed; an external agent moves q slowly outward with negligible change in kinetic energy or other transfers.

Doubling radius lowers positive potential but raises the negative pair energyThe source Q is centred at drawing x fifty. Alternative states one and two of the negative test charge lie at x 170 and 290 on the same horizontal line. Thus their source separations are in the exact one-to-two ratio, representing 0.300 and 0.600 metres. A blue prescribed-movement arrow points from state one towards state two. State one is dashed and state two solid to distinguish alternatives rather than two simultaneous test charges. The source potential decreases from positive 120 to positive sixty volts. Pair potential energy rises from negative 2.40 to negative 1.20 times ten to the minus seven joules. The positive energy change is 1.20 times ten to the minus seven joules; the field does negative work during this slow outward movement. Zero potential and zero pair energy at infinity are reference limits, not an endpoint shown in this finite distance drawing.Prescribed slow outward moveState 1State 2+--Qfixed0.300 m0.600 mV: +120 V to +60.0 VUE / 10-7 J: -2.40 to -1.20

With V = 0 and UE = 0 at infinity, ΔUE = qΔV = +1.20 × 10-7 J. External work is positive; electric-field work is negative. Potential in volts and pair energy in joules are different quantities.

The two radii are alternative positions of one negative charge, not two simultaneous test charges. The prescribed movement is slow and outward. Zero potential and pair energy at infinity are limiting references, not finite endpoints.
ΔUE = (-2.00 × 10-9)(60.0 - 120)
= +1.20 × 10-7 J

Potential falls, but the negative charge's potential energy rises. With negligible kinetic-energy change and other transfers, the external work is +1.20 × 10-7 J; the electric field does -1.20 × 10-7 J of work.

Optional check A -2.00 nC charge is moved slowly from potential +120 V to +60.0 V in a fixed source field. With no kinetic-energy change, what is the change in the pair's electric potential energy?
A -2.00 nC charge is moved slowly from potential +120 V to +60.0 V in a fixed source field. With no kinetic-energy change, what is the change in the pair's electric potential energy?

If the field alone changes a particle's kinetic energy in the fixed-source model, neglecting radiation and other transfers, then ΔEk = -ΔUE = -qΔV. State whether motion is externally controlled or driven by the field alone before assigning work or a speed.