Prof. David R. Jackson ECE Dept.

ECE 2317 Applied Electricity and Magnetism Spring 2014 Prof. David R. Jackson ECE Dept. Notes 4 Notes prepared by the EM Group University of Houston

Electric Field + + + + + + - - - - - - + - + + + - - + - - - + + • V0 [V] E q - Note: The “test” charge is small enough so that it does not disturb the charge on the capacitor and hence the field from the capacitor. so The electric-field vector is the force vector for a unit charge. Units of electric field: [V/m]

Electric Field (cont.) Note: The electric-field vector may be non-uniform. E Point charge q in free space: q

Voltage Drop The voltage drop between two points is now defined. B Volta C A E(x,y,z) Comment: In statics, the line integral is independent of the shape of the path. (This will be proven later after we talk about the curl.)

Voltage Drop (Cont.) We now explore the physical interpretation of voltage drop. B q test charge A C E(x,y,z) Fext(x,y,z) A test charge is moved from point A to point B at a constant speed (no increase in kinetic energy).

Voltage Drop (cont.) B q test charge A C E(x,y,z) Fext(x,y,z) Wext = work done by observer in moving the charge. So

Voltage Drop (cont.) But we also have so or

Physical Interpretation of Voltage + - + - - + - + + - - + + + + + + + - - - - - - The voltage drop is equal to the change in potential energy of a unit test charge. Point A is at a higher voltage, and hence a higher potential energy, than point B. A + • V0 [V] E q - B

Comments - + + + + - - + - + - - + - + + + + - - - - - + • The electric field vector points from positive charges to negative charges. • Positive charges are at a higher voltage, and negative charges are at a lower voltage. + • V0 [V] E q -

Review of Doing Line Integrals In rectangular coordinates, Hence Note: The limits are vector points. so Each integrand must be parameterized in terms of the respective integration variable. This requires knowledge of the path C.

Example + + + + + - - - - - - + - - + + + - - + - - + + A x= 0 + E V0 [V] - x=h B Find: E Ideal parallel-plate capacitor assumption. Assume:

Example (cont.) Evaluate in rectangular coordinates:

Example (cont.) Recall that Hence, we have

Example + + + - + + - - - - - + + + + - + + - - - - + - x= 0 x = 0 A + Proton E V0 [V] h= 0.1 [m] - B x = x x=h V0= 9 [V] A proton is released at point A on the top plate with zero velocity. Find the velocity v(x) of the proton at distance x from the top plate (at point B). Conservation of energy:

Example (cont.) Hence:

Example (cont.) V0= 9 [V] h= 0.1 [m] q =1.602 x 10-19 [C] (See Appendix B of the Hayt & Buck book or Appendix D of the Shen& Kong book.) m =1.673 x 10-27 [kg] Hence:

Potential (Voltage) Function R r= (x,y,z) C (The word “potential” is often used when we talk about the voltage at a single point in space.) Pick a reference pointRwhere the potential is assumed (defined): Recall: or Note: 0 is often chosen to be zero. so

Example - + 9 [V] B A Find the potential on the left terminal (cathode) assuming R is on right terminal (anode) and = 0 at the reference point. B A 9 [V]

Note on Grounding B A - + 9 [V] Note: Choosing a reference point is NOT the same thing as grounding. • Grounding means connecting to the physical earth. • It is done for safety and to reduce noise. Ground does not have to a reference point. We still have: To illustrate: Reference point Ground

Note on Grounding (cont.) B A 9 [V] Reference point - + Ground • This is the most common scenario: • Ground is chosen to be a reference point • The reference voltage is chosen to be zero. • Notes: • In DC systems, the negative terminal is usually grounded. • In coaxial cable systems, the outer conductor (“shield”) is usually grounded.

Example - - + - + + - + + - + - - - - - - - + + + + + + x= 0 + r(x, y, z) E V0 [V] - x=h Find the potential function at x,assuming that the reference point R is on the bottom plate, and the voltage at R is zero.

Prof. David R. Jackson ECE Dept.