Electric Potential & Voltage

What is Electric Potential & Voltage?

Electric potential V at a point in space is the work per unit positive charge required to bring a test charge quasi-statically from infinity to that point: V = kq/r for a single point charge (in joules per coulomb, or volts). Unlike the electric field — which is a vector quantity requiring geometric addition — potential is a scalar, so contributions from multiple charges sum algebraically: V_total = k∑(qᵢ/rᵢ). Equipotential lines connect points of equal V; they are always perpendicular to electric field lines because no work is done moving a charge along them. This simulation maps the full potential landscape as a color gradient, overlays equipotential contour lines, and lets you place one to three point charges to observe how their scalar fields superpose and how the resulting E field — obtained from E = −∇V — relates geometrically to the contours you see.

Parameters explained

Source Charge(μC)

The magnitude and sign of the primary source charge in microcoulombs (μC). Positive values produce a hill-shaped potential centered on the charge (high V nearby, decreasing outward); negative values create a potential well (low V nearby). The color map and contour spacing rescale automatically to the range set by this parameter.

Test Charge Sign (0=Positive, 1=Negative)

Sets the sign of the test charge used to compute work and potential energy overlays: 0 = positive test charge, 1 = negative test charge. The potential V at a point is independent of the test charge sign, but the potential energy U = qV and the direction of spontaneous motion (from high to low potential energy) depend on it. Toggle this to illustrate that positive charges accelerate toward decreasing V while negative charges accelerate toward increasing V.

Number of Charges

Number of source charges rendered simultaneously (1 to 3). Adding a second or third charge demonstrates superposition: V_total is the algebraic sum of individual potentials. With charge_count = 2, the potential saddle point between like charges and the equipotential distortion near opposite charges become visible.

Charge 2 X Position(m)

Horizontal position of the second source charge in meters, measured from the center of the canvas. Sliding this parameter changes the separation between charges and reshapes the equipotential pattern — moving like charges apart spreads the saddle point; moving opposite charges closer pinches the equipotential lines between them. Active only when charge_count ≥ 2.

Common misconceptions

• MisconceptionVoltage and electric field are the same thing — a high-voltage region has a strong electric field.

  CorrectV (volts, J/C) is potential energy per unit charge at a point; E (N/C or V/m) is force per unit charge. The two are related by E = −dV/dr (or more generally E = −∇V). A region can have high potential but weak field — for example, inside a large hollow conductor (or a conducting sphere in electrostatic equilibrium) where V is large but E = 0.

• MisconceptionEquipotential lines show where the electric field is strongest — they bunch together near high field.

  CorrectClosely spaced equipotentials indicate a large potential gradient, which means a large field magnitude |E| = |dV/dr|. But the equipotential lines themselves are perpendicular to E, not parallel. The field points from high-V to low-V regions, crossing each contour at right angles.

• MisconceptionMoving a charge along an equipotential line requires work because you are moving against the electric field.

  CorrectBy definition, all points on an equipotential surface have the same V, so ΔV = 0. The electric field E is perpendicular to the equipotential by construction, so it exerts no tangential force on a charge moving along the surface. Therefore W = qΔV = 0: no work is done by the field (or against it) for any path that stays on a single equipotential.

• MisconceptionElectric potential is always positive.

  CorrectV = kq/r is negative for a negative source charge q, and can be negative at any point where the negative contributions outweigh positive ones. The simulation's color map uses cool colors for negative V and warm colors for positive V; the zero-potential equipotential is the transition between them and is visible as a distinct contour.

• MisconceptionThe electric field points from low potential to high potential, toward positive charges.

  CorrectThe field points from high potential to low potential: E = −∇V. Near a positive charge, V is highest close to the charge and decreases outward, so E points outward (away from the charge). Near a negative charge, V is most negative at the source, so E points inward toward the charge.

How teachers use this lab

1. Potential vs. distance data collection: set charge_count = 1, charge_value = 5 μC. Have students click the canvas at radii of 0.5, 1.0, 1.5, 2.0, and 2.5 m from the charge to read V at each point. They compute kq/r analytically and compare. Plotting V vs. 1/r should yield a straight line, confirming the 1/r dependence. Addresses AP standard CHA-2.C.
2. Superposition demonstration: set charge_count = 2, charge_value = +5 μC, and slide charge2_x from −2 m to +2 m. Have students identify the stationary point where E = 0 between the two like charges (V reaches a local minimum along the line between them, but never equals zero for two positive charges) and trace how it shifts with separation. Connecting V_total = k(q₁/r₁ + q₂/r₂) to the visual makes superposition concrete. Addresses standard CHA-2.D.
3. Perpendicularity of E and equipotentials: after setting charge_count = 1, have students draw freehand arrows on a printed screenshot of the equipotential map representing their guess for the E field direction at five points. Then overlay the actual E vectors from the simulation and measure the angle between E and the nearest equipotential. Students discover the 90° relationship and connect it to E = −∇V. Addresses standard CHA-3.A.
4. Misconception probe — does voltage equal field strength?: set charge_value to +10 μC and identify a point very close to the charge (high V and high E) and a point on a large-radius equipotential (lower V but compare E values). Ask students whether the higher-V region always has higher E. Use the simulation's V and E readouts to show that inside a 3D region where V is uniform — for example the interior of a conductor in electrostatic equilibrium — E = 0; on a 2D equipotential surface around an isolated charge, V is constant along the surface but E is nonzero and perpendicular to it.
5. Work calculation from ΔV: set charge_count = 1 and charge_value = 5 μC. Ask students to compute the work for a +1 nC test charge moved from V_i (at r = 2 m) to V_f (at r = 0.5 m). Use the unified convention: ΔU = qΔV = q(V_f − V_i); work done by the field W_field = −ΔU = q(V_i − V_f); work done by an external agent quasi-statically W_ext = +ΔU = q(V_f − V_i). Students confirm that moving a positive charge toward higher V requires positive external work. Addresses standard CHA-2.D.