Prof. David R. Jackson ECE Dept. Fall 2014 Notes 4 ECE 2317 Applied Electricity and Magnetism Notes prepared by the EM Group University of Houston 1

so Electric Field The electric-field vector is the force vector for a unit charge. Units of electric field: [ V/m ] V 0 [ 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. 2

Electric Field (cont.) Note: The electric-field vector may be non-uniform. E q Point charge q in free space: 3

Voltage Drop The voltage drop between two points is now defined. Volta 4 C A B 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. C A B E (x,y,z) q test charge F ext (x,y,z) A test charge is moved from point A to point B at a constant speed (no increase in kinetic energy). 5

Voltage Drop (cont.) W ext = work done by observer in moving the charge. 6 C A B E (x,y,z) q test charge F ext (x,y,z) So

Voltage Drop (cont.) so But we also have or 7

Physical Interpretation of Voltage 8 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 V 0 [ V ] E q = 1 A B

Comments 9  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 V 0 [ V ] E

Review of Doing Line Integrals 10 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 Find: E Assume: V 0 [ V ] E x = 0 x = hx = h A B 11 Ideal parallel-plate capacitor assumption.

Example (cont.) Evaluate in rectangular coordinates: 12

Example (cont.) Hence, we have Recall that 13

Example 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). 14 Conservation of energy: h = 0.1 [ m ] V 0 = 9 [ V ] V 0 [ V ] E x = 0 x = hx = h Proton A B x = x x = 0

Example (cont.) Hence: 15

Example (cont.) Hence: h = 0.1 [ m ] V 0 = 9 [ V ] q = x [C] m = x [kg] (See Appendix B of the Hayt & Buck book or Appendix D of the Shen & Kong book.) 16

Reference Point  A reference point R is a point where the voltage is assigned.  This makes the voltage unique at all points.  The voltage at a given point is often called the potential . R C r = (x,y,z) Note:  0 is often chosen to be zero, but this is not necessary. 17 Integrating the electric field along C will determine the potential at point r.

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

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