Topic 9.3 Electric Field, Potential, and Energy (2 hours)

Recall the definition of Electric Potential We define the electric potential (V) at a point as the work per unit charge that must be done to bring a small positive test charge from far away to the point of interest. where V is the potential in volts (1 V = 1 J/C), W is the work done (in J) which equals the potential energy, and q is the magnitude of the test charge (in C).

Charges and Potential Energy How much work must be done to bring in that small positive test charge from infinity? q1 q2 q2

And because the work done is equal to the potential energy at that point we have and the electric potential caused by a point charge is NOTE: See Data Booklet

Example: Determine the electric potential on the surface of a gold nucleus that has a radius of 6.2 fm.

Electric Potential Energy and Electric Potential Note that electric potential energy and electric potential are scalar quantities (they can be positive or negative, but they have no direction). Example: Find the electric potential energy for four charges (2.0 mC, 3.0 mC, 4.0 mC, 5.0 mC) placed at the vertices of a square of side 10.0 cm. Now find the electric potential at the center of the square.

Electric Field and Electric Potential Consider the movement of a small positive test charge q in a uniform electric field as shown below.

Electric Field and Electric Potential Suppose again that the charge +q is moved a small distance by a force F from A to B so that the force can be considered constant. The work done is given by: ΔW = F Δx

Electric Field and Electric Potential The force F and the electric field E are oppositely directed, and we know that: F = -qE and ΔW = q ΔV = F Δx Therefore, the work done can be given as: q ΔV = -q E Δx Therefore Electric field is the negative gradient of potential.

Example: Determine how far apart two parallel plates must be situated so that a potential difference of 150. V produces an electric field strength of 1.00 × 103 NC-1.

Electric Field and Electric Potential of a Charged Sphere of Radius r0 Outside the sphere, the graphs are the same as those of a point charge. At the surface, r = r0. Therefore, field and potential are at their maximum values. Inside the sphere, the electric field is zero. Inside the sphere, no work is done to move a charge from a point inside to the surface. Therefore, there is no potential difference and the potential is the same as it is when r = r0.

Equipotentials Regions in space where the electric potential of a charge distribution has a constant value are called equipotentials. The places where the potential is constant in three dimensions are called equipotential surfaces, and where they are constant in two dimensions they are called equipotential lines. They are in some ways analogous to the contour lines on topographic maps where the gravitational potential energy is constant as a mass moves around the contour line because the mass remains at the same elevation above the Earth’s surface. The gravitational field strength acts in a direction perpendicular to a contour line.

Field Lines and Equipotentials Similarly, because the electric potential on an equipotential line has the same value, no work can be done by an electric force when a test charge moves on an equipotential. Therefore, the electric field cannot have a component along an equipotential, and thus the electric field must be everywhere perpendicular to the equipotential surface or equipotential line. This fact makes it easy to plot equipotentials if the lines of force or electric field lines are known.

Field Lines and Equipotentials NOTE: All points on the surface of a charged conductor are at the same potential. That is, its surface is an equipotential surface.

Example: Two spheres of radii r and R = 10r are connected by a long conducting wire. Before connecting, the big sphere had an amount of charge Q on it and the smaller sphere was uncharged. How much charge is there on each sphere now?