Topic 4 of 6
Field work and potential energy
Potential energy belongs to an interaction or configuration. Determine the field force and displacement before deciding whether that energy increases or decreases.
- Gravitational potential energy
- Associated with the relative positions of masses, such as a load and Earth. It is not restricted to a height model in every gravitational situation.
- Electric potential energy
- Associated with the positions of interacting charges. The test charge's sign matters when relating a movement to its potential-energy change.
- Elastic potential energy
- Associated with deformation of a material, such as stretching or compressing a spring. Its change is related to the force-extension area.
Choose a reference when giving an absolute potential-energy value. Adding the same constant to the initial and final values does not change their difference. A change can therefore be calculated even when the chosen zero is elsewhere.
Work by a static field
For the gravitational and electric fields considered here:
Positive work by the field decreases the associated potential energy. Negative field work increases it. This statement applies to the field's work; it does not say that every force's work equals minus a potential-energy change.
For gravity, the force on a positive mass is along the gravitational field. In an electric field, F = qE: the force on a negative charge is opposite the field arrow. Calculate work using the actual force direction.
Compare the field force with the prescribed displacement
Green shows field direction, purple the force exerted by that field and blue the movement. These panels isolate field work; they are not complete free-body diagrams. Right or up is positive as shown.
Lift a mass against the gravitational field
Gravity does -3.924 J of work; the body-Earth gravitational potential-energy change is +3.924 J.
Positive charge: field force and movement agree
Field work is +3.0 × 10-4 J, so electric potential-energy change is -3.0 × 10-4 J.
Negative charge: the field itself does not reverse
Field work is -3.0 × 10-4 J, so electric potential-energy change is +3.0 × 10-4 J. A prescribed rightward displacement does not imply rightward acceleration.
Move a mass upward
A 0.500 kg body is moved 0.800 m upward in a uniform gravitational field of 9.81 N/kg downward. Choose upward as positive. Gravitational force is 0.500 × 9.81 = 4.905 N downward, or -4.905 N in this coordinate.
ΔEp = +3.924 J ≈ +3.92 J
Gravity opposes the upward displacement, so the load-Earth gravitational store increases. Reversing the displacement reverses the two signs: gravity then does positive work and the gravitational potential energy decreases.
Compare positive and negative charges
Choose right as positive. In a uniform electric field E = 2.0 × 103 N/C right, move a charge 0.050 m right. Recall that 1 microC = 10-6 C.
A +3.0 microC charge
Wfield = (+0.0060)(+0.050) = +3.0 × 10-4 J
ΔEp = -3.0 × 10-4 J
The force and displacement point right, so the field does positive work. Electric potential energy decreases. That energy may increase kinetic energy or transfer out through another force, depending on how the body moves.
A -3.0 microC charge in the same field
Wfield = (-0.0060)(+0.050) = -3.0 × 10-4 J
ΔEp = +3.0 × 10-4 J
The charge is still moved right, but its force is left. The field does negative work and electric potential energy increases. Changing the test charge's sign has not reversed the source field.
Distinguish field work from an agent's work
If an external agent moves the body slowly with unchanged kinetic energy and no other energy transfers, its work is +ΔEp. Thus steadily lifting the mass requires +3.924 J from the agent, and steadily moving the negative charge right requires +3.0 × 10-4 J.
If the body also gains kinetic energy, or energy dissipates, the agent's total work need not equal the potential-energy increase alone. Write the complete energy account rather than assigning every work term to one store.