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Electric Potential and Energy Physics 201 Professor P. Q. Hung 311B, Physics Building Physics 201 – p. 1/28 Electric Potential and Energy Summary of last lecture Electric Potential for a constant electric field: VB − VA = −E.~s~ Physics 201 – p. 2/28 Electric Potential and Energy Summary of last lecture Electric Potential for a constant electric field: VB − VA = −E.~s~ Electric Potential for a point charge: q V = k r Physics 201 – p. 2/28 Electric Potential and Energy Equipotential surfaces and Electric fields: Example A parallel plate capacitor is composed on two charged plates, one positive and the other one negative. A constant electric field points from the positive plate to the negative plate. Suppose the potential difference between the 2 plates is 64 V and that they are separated by 0.032m. By this we mean ∆V = V− − V+ = −64V . Between the two plates, one can draw equipotential planes parallel to the plates. What is the separation between 2 of such planes if 3.0 V ? Physics 201 – p. 3/28 Electric Potential and Energy Equipotential surfaces and Electric fields Equipotential Surface: Surface where the electric potential is the same everywhere. Physics 201 – p. 4/28 Electric Potential and Energy Equipotential surfaces and Electric fields Equipotential Surface: Surface where the electric potential is the same everywhere. What are the equipotential surfaces around a point charge? Physics 201 – p. 4/28 Electric Potential and Energy Equipotential surfaces and Electric fields Equipotential Surface: Surface where the electric potential is the same everywhere. What are the equipotential surfaces around a point charge? What are the equipotential surfaces between two charged parallel plates (one + and one -) (parallel-plate capacitor)? Physics 201 – p. 4/28 Electric Potential and Energy Equipotential surfaces for a point charge q V = k r Physics 201 – p. 5/28 Electric Potential and Energy Equipotential surfaces for two point charges q V = k r Physics 201 – p. 6/28 Electric Potential and Energy Equipotential surfaces for a parallel-plate capacitor Recall Active Example 19.3: ~ σ : Constant |E| = ǫ0 electric field. Physics 201 – p. 7/28 Electric Potential and Energy Equipotential surfaces and Electric fields: Meanings When a particle moves from one point to another point on a equipotential surface, the net electric force does no work. Since VB − VA = −WAB/q, “equipotential” means that VB = VA ⇒ WAB = 0. Physics 201 – p. 8/28 Electric Potential and Energy Equipotential surfaces and Electric fields: Meanings The electric field always point in the direction perpendicular to the equipotential surface. If not, it would have a component parallel to the surface which means that the surface is no longer equipotential since the electric field points in the direction of decreasing potential Physics 201 – p. 9/28 Electric Potential and Energy Equipotential surfaces and Electric fields: Meanings A conductor’s surface is an equipotential surface because the electric field has to be perpendicular to the conductor’s surface. Since there cannot be an electric field inside the conductor, the electric potential inside has the same value as that on the surface. Physics 201 – p. 10/28 Electric Potential and Energy Charged conductor of arbitrary shape Charged conducting sphere: 1) V = 4πkσR on the surface and inside; 2) E = 4πkσ on the surface and perpendicular to it and zero inside. Physics 201 – p. 11/28 Electric Potential and Energy Charged conductor of arbitrary shape Charged conducting sphere: 1) V = 4πkσR on the surface and inside; 2) E = 4πkσ on the surface and perpendicular to it and zero inside. Two spheres with different radii, R1 and R2 and same potential ⇒ σ1R1 = σ2R2 ⇒ σ1 = σ2R2/R1 Physics 201 – p. 11/28 Electric Potential and Energy Charged conductor of arbitrary shape Charged conducting sphere: 1) V = 4πkσR on the surface and inside; 2) E = 4πkσ on the surface and perpendicular to it and zero inside. Two spheres with different radii, R1 and R2 and same potential ⇒ σ1R1 = σ2R2 ⇒ σ1 = σ2R2/R1 If R1 ≪ R2 ⇒ σ1 ≫ σ2 ⇒ E1 ≫ E2 ⇒ Sharp ends of a charged conductor have larger electric fields. Physics 201 – p. 11/28 Electric Potential and Energy Charged conductor of arbitrary shape Physics 201 – p. 12/28 Electric Potential and Energy Equipotential surfaces and Electric fields: Medical applications The body is not an ideal conductor ⇒ differences in potential from one place to another Physics 201 – p. 13/28 Electric Potential and Energy Equipotential surfaces and Electric fields: Medical applications The body is not an ideal conductor ⇒ differences in potential from one place to another Differences in potential ⇒ Electrocardiograph and electroencephalograph Physics 201 – p. 13/28 Electric Potential and Energy Equipotential surfaces and Electric fields: Moving from one surface to another Take two equipotential surfaces which are very close to each other so that the electric field is more or less constant: ∆ V ∆ V = −E∆ s ⇒ E = − ∆ s Physics 201 – p. 14/28 Electric Potential and Energy Equipotential surfaces and Electric fields: Moving from one surface to another Take two equipotential surfaces which are very close to each other so that the electric field is more or less constant: ∆ V ∆ V = −E∆ s ⇒ E = − ∆ s ∆ V ∆ s is a potential gradient. The minus sign says that the electric field points in the direction of decreasing potential. Physics 201 – p. 14/28 Electric Potential and Energy Equipotential surfaces and Electric fields: Moving from one surface to another Take two equipotential surfaces which are very close to each other so that the electric field is more or less constant: ∆ V ∆ V = −E∆ s ⇒ E = − ∆ s ∆ V ∆ s is a potential gradient. The minus sign says that the electric field points in the direction of decreasing potential. Change in the electric potential ⇒ electric field. Physics 201 – p. 14/28 Electric Potential and Energy Equipotential surfaces and Electric fields: Example A parallel plate capacitor is composed on two charged plates, one positive and the other one negative. A constant electric field points from the positive plate to the negative plate. Suppose the potential difference between the 2 plates is 64 V and that they are separated by 0.032m. By this we mean ∆V = V− − V+ = −64V . Between the two plates, one can draw equipotential planes parallel to the plates. What is the separation between 2 of such planes if 3.0 V ? Physics 201 – p. 15/28 Electric Potential and Energy Equipotential surfaces and Electric fields: Solution to Example Between 2 parallel charged plates, the electric field is constant. One obtains ∆V −64 V E = − ∆s = −0.032m = 2.0 × 103 V/m The electric field is pointing in the direction of decreasing potential. Physics 201 – p. 16/28 Electric Potential and Energy Equipotential surfaces and Electric fields: Solution to Example Let ∆d be the separation between these two equipotential surfaces. Since E is constant, one has ∆V −3.0V ∆d = − E = −2.0×103 V/m = 1.5 × 10−3m Physics 201 – p. 17/28 Electric Potential and Energy Capacitors Example of a capacitor: two oppositely-charged parallel plates . Physics 201 – p. 18/28 Electric Potential and Energy Capacitors Example of a capacitor: two oppositely-charged parallel plates . Experiment: Increase the charge Q on each plate ⇒ Direct increase in the potential difference. ⇒ Q is directly proportional to ∆V . Physics 201 – p. 18/28 Electric Potential and Energy Capacitors The constant of proportionality is called the capacitance, i.e. the capacity of the device to store charge. Q = C|∆V | (6) C is called the capacitance (a positive number always). Unit: 1 farad(F ) = 1 C/V . Physics 201 – p. 19/28 Electric Potential and Energy Capacitors Numerous applications: For example, a RAM chip consists of millions of transitor-capacitor units. Capacitor is charged: 1. Capacitor is uncharged: 0. Physics 201 – p. 20/28 Electric Potential and Energy Capacitors: Calculating capacitances Parallel-plate capacitor of separation d and area A: Three pieces of information: E = σ/ǫ0 = Q/(ǫ0A), |∆V | = Ed, and C = Q/|∆V |.⇒ Physics 201 – p. 21/28 Electric Potential and Energy Capacitors: Calculating capacitances Parallel-plate capacitor of separation d and area A: Three pieces of information: E = σ/ǫ0 = Q/(ǫ0A), |∆V | = Ed, and C = Q/|∆V |.⇒ Q ǫ0A C = Ed = d Purely geometrical! Physics 201 – p. 21/28 Electric Potential and Energy Capacitors: Example When a potential difference of 150 V is applied to the plates of a parallel- plate capacitor, the plates carry a surface charge density of 30.0 nC/cm2. What is the spacing between the plates? ǫ0A ǫ0A Use C = d . ⇒ d = C . Physics 201 – p. 22/28 Electric Potential and Energy Capacitors: Example When a potential difference of 150 V is applied to the plates of a parallel- plate capacitor, the plates carry a surface charge density of 30.0 nC/cm2. What is the spacing between the plates? ǫ0A ǫ0A Use C = d . ⇒ d = C . C = Q/|∆V | = σA/|∆V | ⇒ ǫ0|∆V | −6 d = σ = 4.42 × 10 m Physics 201 – p. 22/28 Electric Potential and Energy Capacitors: Dielectric Electric dipole moments: centers of positive and negative charges do not coincide. Some material have permanent electric dipole moments Physics 201 – p. 23/28 Electric Potential and Energy Capacitors: Dielectric Electric dipole moments: centers of positive and negative charges do not coincide. Some material have permanent electric dipole moments Insert a slab of such material in between two oppositely charged plates. Physics 201 – p. 23/28 Electric Potential and Energy Capacitors: Dielectric Between the plates, the negative sides are attracted to the positive plate and the positive sides are attracted to the negative plate, creating an electric field which points in the opposite direction to that of the external electric field E0, and partially cancelling it inside.
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