Electrostatics: Electric Potential

Size: px

Start display at page:

Download "Electrostatics: Electric Potential"

Angelica Clark
5 years ago
Views:

1 Electrostatics: EE3321 Electromagnetic Field Theory Outline Concept of electric potential V Potential difference V Electric potential due to charge Relationship between E and V Electric potential example Electrostatics -- Slide 2 1

2 Concept of Vector Calculus Derivation of Recall from vector calculus that the gradient of a scalar function cannot have any curl. V 0 Recall for electrostatics that the electric field has zero curl. E 0 This means that the electric field intensity can be written as the gradient of a scalar function V. E V negative sign is incorporated to enforce the sign convention that electric fields are directed from high to low potential. Electrostatics -- Slide 4 2

3 Intuitive Derivation of Electric Potential Suppose a voltage V 0 is applied across two plates bounding some medium. z V d Vd V0 V 0 0 Potential will vary linearly between the plates. Electrostatics -- Slide 5 Intuitive Derivation of Electric Potential This is induces charge in the plates at either side of the medium. z V 0 - d Vd V0 V 0 0 Electrostatics -- Slide 6 3

4 Intuitive Derivation of Electric Potential The charge creates an electric field between the plates that is known to be uniform. z V 0 - d Vd V0 V 0 0 Electrostatics -- Slide 7 Intuitive Derivation of Electric Potential Conclusion the electric field is the slope of the voltage. V E z a ˆz z V 0 - d Vd V0 V 0 0 Electrostatics -- Slide 8 4

5 Intuitive Derivation of Electric Potential This can be generalized to 3D using the gradient operation. E V z a ˆz E V This is the equation used to calculate the electric field from the electric potential. Electrostatics -- Slide 9 Potential Difference 5

6 Work to Move a Charge Suppose a point charge is moved from point to point, a distance of d, in the presence of an electric field E. Force on the charge F E d E Work done to move charge W Fd d E The negative sign indicates the force is external. This can be generalized to a differential distance dw FdE d Integrate to get total work. W dw Ed Electrostatics -- Slide 11 Potential Difference The potential difference between points and is the potential energy per unit charge. Potential energy is the same as the work it would take to move the charge. d E Divide total work by W Ed W E d This is the potential difference between points and. W V V V Ed Electrostatics -- Slide 12 6

7 One pplication of Potential Difference The equation below is used to calculate the electric potential from the electric field. V V V Ed Electrostatics -- Slide 13 Notes bout Potential Difference is the initial point and is the final point. This is important for the sign convention. V < 0 indicates a loss in potential energy because work is being done by the field. V > 0 indicates a gain in potential energy because an external agent must be doing the work. V is independent of the path taken from to. V is measured in joules per Coulomb (J/C), or volts (V). Electrostatics -- Slide 14 7

8 Due to Charge Recall Electric Field round a Charge Electric flux density D E ar ˆ 2 4 R Electric Field Intensity D 4 R ˆ 2 a R Electrostatics -- Slide 16 8

V rv ˆ ˆ ref a 2 R dra 4 r r 1 dr 2 4 r R 1 4 r V r r 1 4 r 1 1 4 r Vref 4 r r R Equation from last slide ring constants to

9 Derivation Setup E r 4 r V r? r a ˆ 2 R Start with the basic equation to calculate V from E. V V Ed For this problem, the equation becomes r VrV ˆ ˆ ref a 2 R dra 4 r R E V 0 Vref Electrostatics -- Slide 17 The Derivation V rv ˆ ˆ ref a 2 R dra 4 r r 1 dr 2 4 r R 1 4 r V r r 1 4 r r Vref 4 r r R Equation from last slide ring constants to outside of integral Calculate anti-derivative Cancel negative signs Evaluate anti-derivative at limits Simplify and bring V ref to right side Electrostatics -- Slide 18 9

Potential Due to Point Charge V r Vref or V r V 4 r 4 r r ref single

Only the potential difference between two points is ever of interest.

Vr + 1 r r Usually, the reference potential V ref is not written explicitly.

Line Charge Sheet Charge Volume Charge s v Charge C Line Charge Density C m

10 Potential Due to Point Charge V r Vref or V r V 4 r 4 r r ref single potential has almost no meaning. Only the potential difference between two points is ever of interest. ny background potential, or reference potential V ref, can be chosen. Vr + 1 r r Usually, the reference potential V ref is not written explicitly. Electrostatics -- Slide 19 Potential Due to Charge Distribution Point Charge Line Charge Sheet Charge Volume Charge s v Charge C Line Charge Density C m Surface Charge Density C m s 2 Volume Charge Density C m v 3 V r r 4 1 V d 4 r r 1 s V ds r r 4 S 1 v V dv 4 r r v Electrostatics -- Slide 20 10

11 Example Example Setup What is the potential difference between these objects? V V V V V Electrostatics -- Slide 22 11

12 Solution (1 of 8) Calculate V from E using a line integration. V V V Ed V V Electrostatics -- Slide 23 Solution (2 of 8) Calculate V from E using a line integration. V V V Ed V For easy integration, a path is chosen where the electric field is always parallel to that path. V This will simplify the dot product to Ed E d Electrostatics -- Slide 24 12

13 Solution (3 of 8) How about this path? V V Electrostatics -- Slide 25 Solution (4 of 8) From the figure, the electric field is almost constant along this path. V V Electrostatics -- Slide 26 13

14 Solution (5 of 8) From the figure, the electric field is almost constant along this path. This let s the integral be approximated as V Ed Ed EL V V Electrostatics -- Slide 27 Solution (6 of 8) To estimate L, the line is moved down to the x-axis. L 1.7 V V Electrostatics -- Slide 28 14

15 Solution (7 of 8) To estimate L, the line is moved down to the x-axis. L 1.7 To estimate E, the color is read from the color bar. E 7.0 V V Electrostatics -- Slide 29 Solution (8 of 8) To estimate L, we move the line down to the x-axis. L 1.7 To estimate E, the color is read from the color bar. E 7.0 V V Finally, the potential difference V is approximately V E L V Electrostatics -- Slide 30 15

16 Interpreting the Sign (1 of 2) Why is the voltage negative? V 11.9 V V V Electrostatics -- Slide 31 Interpreting the Sign (2 of 2) Why is the voltage negative? V 11.9 V Recall the definition of V. V V V Ed From the figure, it is concluded that V > V Therefore V = V V is negative. V V Electrostatics -- Slide 32 16

Electrostatics: Electrostatic Devices

Electrostatics: Electrostatic Devices EE331 Electromagnetic Field Theory Outline Laplace s Equation Derivation Meaning Solving Laplace s equation Resistors Capacitors Electrostatics -- Devices Slide 1