Physics 2112 Unit 6: Electric Potential Today’s Concept: Electric Potential (Defined in terms of Path Integral of Electric Field)

Big Idea Last time we defined the electric potential energy of charge q in an electric field: The only mention of the particle was through its charge q. We can obtain a new quantity, the electric potential, which is a PROPERTY OF THE SPACE, as the potential energy per unit charge. The purpose of this Check is to jog the students minds back to when they studied work and potential energy in their intro mechanics class. Note the similarity to the definition of another quantity which is also a PROPERTY OF THE SPACE, the electric field.

Example 6.2a (Potential from Field) A +8uC charge is placed in a downwards pointing electric field with a magnitude of 6N/C. An outside force moves the charge up a distance of 0.4m from point 1 to point 2. = 6 N/C q=+8uC a) What is the force on this charge? b) How much work was done by the outside force during this move? The purpose of this Check is to jog the students minds back to when they studied work and potential energy in their intro mechanics class.

Example 6.2.b (Potential from Field) A +8uC charge is placed in a downwards pointing electric field with a magnitude of 6N/C. An outside force moves the charge up a distance of 0.4m from point 1 to point 2. = 6 N/C q=+8uC c) How much work was done by the electric field during this move? d) What is the change in the electrical potential energy of the particle? The purpose of this Check is to jog the students minds back to when they studied work and potential energy in their intro mechanics class.

Example 6.2.c (Potential from Field) A +8uC charge is placed in a downwards pointing electric field with a magnitude of 6N/C. An outside force moves the charge up a distance of 0.4m from point 1 to point 2. = 6 N/C q=+8uC e) What is the difference in electrical potential between points 1 and 2? f) What is the electrical potential at point 2? The purpose of this Check is to jog the students minds back to when they studied work and potential energy in their intro mechanics class.

Electric Potential from E field Consider the three points A, B, and C located in a region of constant electric field as shown. D Dx What is the sign of DVAC = VC - VA ? A) DVAC < 0 B) DVAC = 0 C) DVAC > 0 The purpose of this Check is to jog the students minds back to when they studied work and potential energy in their intro mechanics class. Choose a path (any will do!)

CheckPoint: Zero Electric Field Suppose the electric field is zero in a certain region of space. Which of the following statements best describes the electric potential in this region? The electric potential is zero everywhere in this region. The electric potential is zero at least one point in this region. The electric potential is constant everywhere in this region. There is not enough information given to distinguish which of the above answers is correct. Remember the definition The purpose of this Check is to jog the students minds back to when they studied work and potential energy in their intro mechanics class.

Example 6.2 (V from point charges) +5nc -5nc A B x -Q +Q 10cm 10cm 10cm What is the electrical potential at points A and B? Define V = 0 at r = (standard)

Example 6.3 (V from point charges) +5nc -5nc 10cm x -Q +Q 10cm 20cm What is the electrical potential at point C? Define V = 0 at r = (standard)

E from V If we can get the potential by integrating the electric field: We should be able to get the electric field by differentiating the potential? In Cartesian coordinates: The purpose of this Check is to jog the students minds back to when they studied work and potential energy in their intro mechanics class.

Example 6.4 (E-field above a ring of charge) What is the electrical potential, V, a distance y above the center of ring of uniform charge Q and radius a? (Assume V = 0 at y = ) P y x y What is the electrical field, E, at that point? a

CheckPoint: Spatial Dependence of Potential 1 The electric potential in a certain region is plotted in the following graph At which point is the magnitude of the E-FIELD greatest? A B C D The purpose of this Check is to jog the students minds back to when they studied work and potential energy in their intro mechanics class.

CheckPoint: Spatial Dependence of Potential 2 The electric potential in a certain region is plotted in the following graph At which point is the direction of the E-field along the negative x-axis? A B C D The purpose of this Check is to jog the students minds back to when they studied work and potential energy in their intro mechanics class.

Example 6.5 (DV near line of charge) An infinitely long solid insulating cylinder of radius a = 4.1 cm is positioned with its symmetry axis along the z-axis as shown. The cylinder is uniformly charged with a charge density ρ = 27.0 μC/m3. Concentric with the cylinder is a cylindrical conducting shell of inner radius b = 19.9 cm, and outer radius c = 21.9 cm. The conducting shell has a linear charge density λ = -0.36μC/m. What is V(P) – V(R), the potential difference between points P and R? Point P is located at (x,y) = (46.0 cm, 46.0 cm). Point R is located at (x,y) = (0 cm, 46.0 cm).

Example 6.5 (DV for charges) cross-section Point charge q at center of concentric conducting spherical shells of radii a1, a2, a3, and a4. The inner shell is uncharged, but the outer shell carries charge Q. What is V as a function of r? a4 a3 +Q a2 a1 +q metal metal Main Idea: Charges q and Q will create an E field throughout space The purpose of this Check is to jog the students minds back to when they studied work and potential energy in their intro mechanics class. Plan: Spherical symmetry: Use Gauss’ Law to calculate E everywhere Integrate E to get V

Equipotentials In previous example, all these points had same V, same electrical potential Line is called “equipotenial” or “a line of equipotential”.

Topographic Map Lines on topo map are lines of “equal gravitational potential” Closer the lines are, the steeper the hill Gravity does no work when you walk along a brown line.

Equipotentials produced by a point charge Equipotentials always perpendicular to field lines. The purpose of this Check is to jog the students minds back to when they studied work and potential energy in their intro mechanics class. SPACING of the equipotentials indicates STRENGTH of the E field.

CheckPoint: Electric Field Lines 1 The field-line representation of the E-field in a certain region in space is shown below. The dashed lines represent equipotential lines. At which point in space is the E-field the weakest? A B C D

CheckPoint: Electric Field Lines 2 The field-line representation of the E-field in a certain region in space is shown below. The dashed lines represent equipotential lines. Compare the work done moving a negative charge from A to B and from C to D. Which one requires more work? More work is required to move a negative charge from A to B than from C to D More work is required to move a negative charge from C to D than from A to B The same amount of work is required to move a negative charge from A to B as to move it from C to D Cannot determine without performing the calculation This CheckPoint can often split the class. So rather than show the results right away, we explain some more, and present a Clicker Question next.