Electric Potential

Electric potential V=Uq is a scalar measured in volts, equivalent to JC. Potential differences determine changes in electric potential energy: ΔU=qΔV. The source fixes V; the charge placed in that potential fixes U. A negative charge changes potential energy in the opposite direction from a positive charge moving through the same potential difference.

The reference zero is a choice. For a finite isolated charge distribution, infinity is a convenient zero. For an ideal infinite line or sheet, potential relative to infinity may not be finite; use a specified reference position instead. Only differences affect the work on a moved charge.

In one dimension E_x=−dV/dx. In several dimensions, each field component is the negative partial derivative with respect to its coordinate. A positive potential slope produces a negative field component. A large potential value does not by itself imply a large field: the local slope is what matters.

Equipotential surfaces are perpendicular to the electric field. Motion along one has ΔV=0, so the electric field does no work. Closely spaced equal potential steps indicate a stronger field. At a point where the potential slope is zero, the field component is zero even if V itself is not.

For point charges and V(∞)=0, V=Σkq_i/r_i. Keep the source charge signs, but do not assign vector directions to the individual potential contributions. Equal opposite charges have zero potential at the midpoint while their fields there reinforce.

For a continuous distribution, V=∫k dq/r. Often this scalar integral is easier than direct field integration; differentiate afterward to obtain a field component. For a ring on its axis, V=kQ√(a²+x²). Differentiation produces the same axial field obtained by resolving every charge element.

The electric force does work W_electric=−ΔU=−qΔV. If the electric force is the only force doing work, ΔK=−qΔV. An external agent moving a charge very slowly does work +ΔU instead. Confusing the work done by the field with work done by the external agent reverses the sign.

For a system of fixed point charges, total configuration energy is Σ over unique pairs kq_iq_j/r_ij. Count every pair once, not twice. Negative total energy means the chosen separated reference has higher energy; it does not mean kinetic energy is negative.

Outside any spherically symmetric finite distribution, V=kQ/r relative to infinity. Inside a conductor in equilibrium, E=0, so V is constant and equals its surface value. Inside a uniformly charged insulating sphere, E is nonzero away from the center, so V continues to change.

Obtain an inside expression by integrating from a known boundary potential. This automatically enforces continuity of V. A finite sheet charge can make the normal field discontinuous at a surface while V stays continuous; distinguish the derivative from the function.

For V=V₀−bx² with b>0, E_x=2bx. A positive charge displaced from zero accelerates farther away; a negative charge accelerates back toward zero. Stability depends on U=qV, so the same potential extremum can be stable for one sign and unstable for the other.

When releasing a charge, use both force and energy: the field gives initial acceleration, while energy gives accessible positions and speed. A charge cannot enter a region requiring negative kinetic energy under the stated energy model.

Electric potential is potential energy per unit charge, measured in joules per coulomb. The source distribution determines V relative to a chosen reference. A test charge has potential energy U=qV, so negative and positive charges can have opposite potential-energy changes through the same potential difference.

Potential is a scalar. Add signed point-charge potentials algebraically without resolving vector components. The field is a vector, so its cancellation conditions differ. At the midpoint between equal positive charges, fields can cancel while the potential is positive relative to infinity. A zero potential at a point does not generally imply zero field.

Only potential differences determine electrostatic work. Choosing zero at infinity is convenient for localized finite distributions, but not always suitable for an ideal infinite distribution. State a reference rather than treating an absolute zero as a universal property.

Potential difference is the negative line integral of E·dl. In a uniform field, displacement along the field lowers potential; perpendicular displacement gives no potential change. In one dimension, Ex=−dV/dx. The negative sign connects field direction to decreasing potential.

On a V-versus-x graph, slope determines the field component. A steep slope means a large field magnitude, while a horizontal segment has zero component in that direction. Curvature alone does not give the field. If a graph crosses V=0 with nonzero slope, the field there is nonzero.

Equipotential surfaces are perpendicular to electrostatic field lines. Closer spacing between surfaces representing equal potential increments indicates a larger field magnitude. Moving along an equipotential changes neither V nor a fixed charge’s electrostatic potential energy, although other forces may still do work.

For motion under electrostatic forces alone, ΔK=−ΔU=−qΔV. A positive charge released from rest accelerates toward lower potential, whereas a negative charge tends toward higher potential because that lowers qV. The field direction is defined using a positive test charge, not the direction every particle must move.

When a particle already has velocity, its trajectory depends on both initial motion and acceleration. Energy relates speed to potential difference but does not by itself determine the full path or elapsed time. Use force and kinematics for trajectory details when needed.

For several interacting point charges, system energy sums each distinct pair once. Counting both q₁q₂ and q₂q₁ double-counts the same interaction. Bringing charges together quasistatically requires external work related to the potential-energy change under the stated assumptions.

For an extended source, dV=k dq/r. Because potential is scalar, this integral can be simpler than a field-component integral. For a uniformly charged ring, every source element has the same distance to an axial point, so V=kQ√(R²+x²) relative to infinity.

Differentiate the resulting potential carefully to obtain the axial field component. This method still requires a correct reference and geometry. It does not justify differentiating with respect to a source coordinate when the desired field concerns the observation coordinate.

Check units and limits. Potential scales as charge divided by distance for a point source, while field scales as charge divided by distance squared. A missing derivative factor often becomes visible through dimensional analysis or failure to approach the far-field result.

Electric potential differences are measurable through work per charge. An absolute potential requires a reference, commonly zero at infinity for a finite localized distribution. For an ideal infinite line or plane, that reference may be unsuitable; choose a finite reference and calculate differences instead.

Potential is a scalar sum: V=k∫dq/r for a finite distribution with the usual zero at infinity. Unlike a field integral, it does not need vector components. A symmetric source can have zero field at a point while its potential there is nonzero. For a charged ring, V=kQ√(R²+x²), and differentiating with a minus sign recovers the axial field.

For several source charges, sum their signed contributions. Negative charges contribute negative potential under the usual reference, but a negative potential does not tell you the field direction. The spatial change in potential, rather than its value alone, determines the field.

The electric force does work W electric=−ΔU=−qΔV. If no other force does work, ΔK=−qΔV. A positive charge released from rest tends to move toward lower potential when free to accelerate along the field; a negative charge gains kinetic energy by moving toward higher potential. The sign of q is essential.

External work in a slow controlled displacement is approximately ΔU when kinetic energy changes negligibly. That differs in sign from the work done by the electric force. State whether the question asks about the field, an external agent, or the total system before choosing a formula.

For a collection of point charges, total interaction energy sums each distinct pair once. Adding qV at every charge using all other charges counts pairs twice unless a factor of one-half is included. A negative total interaction energy can represent an arrangement requiring positive work to separate to infinity; it is not negative kinetic energy.