Dr. Hugh Blanton ENTC 3331

Energy & Potential

Dr. Blanton - ENTC Energy & Potential 3 The work done, or energy expended, in moving any object a vector differential distance d l under the influence of a force is: Work, or energy, is measured in joules (J).

Dr. Blanton - ENTC Energy & Potential 4 The differential electric potential energy dW per unit charge is called the differential electric potential (or differential voltage) dV. Convention: potential at infinity  0  ground. minus sign is convention

Dr. Blanton - ENTC Energy & Potential 5 Electrical potential of a point charge & E-field of a point charge, q, located at the origin?

Dr. Blanton - ENTC Energy & Potential 6 Compare the electric field and potential of a point charge. Expressions are rather similar.

Dr. Blanton - ENTC Energy & Potential 7 Refer to the gradient in the spherical coordinates no dependence on  or 

Dr. Blanton - ENTC Energy & Potential 8 It follows that for a point charge:

Dr. Blanton - ENTC Energy & Potential 9 Generalizations: All complicated charge distributions are always superposition of fields due to many point charges. All these fields superimpose linearly.

Dr. Blanton - ENTC Energy & Potential 10 Multiple charges Charge distributions

Dr. Blanton - ENTC Energy & Potential 11 And importantly, for all distributions. Note potential energy {do not confuse with potential energy} Coulomb force

Dr. Blanton - ENTC Energy & Potential 12 From mechanics For both Coulomb and mechanical forces, the force is equal to the gradient of the potential energy. Gravitational force potential energy

Dr. Blanton - ENTC Energy & Potential 13 This isn’t so surprising, since Coulomb’s law And the gravitational force are so similar.

Dr. Blanton - ENTC Energy & Potential 14 Forces for which holds are conservative. That is, the energy associated with the conservative force is conserved.

Dr. Blanton - ENTC Energy & Potential 15 Determine the electric potential, V, at the origin for the figure. For multiple charges: For all four charges: r is the origin

Dr. Blanton - ENTC Energy & Potential 16 Important observation while

Dr. Blanton - ENTC Energy & Potential 17 Find the E-field above a circular disk of radius a with  S = constant. What are the potential, V, and E-field at point P(0,0,z) right above the circular disk? y z x

Dr. Blanton - ENTC Energy & Potential 18 y z x

Dr. Blanton - ENTC Energy & Potential 19 The question is, what is the field

Dr. Blanton - ENTC Energy & Potential 20 Since

Dr. Blanton - ENTC Energy & Potential 21 The field points along the z-axis: This result is identical to that obtained earlier. By first calculating the potential V and then using, considerable geometrical considerations can be avoided.

Dr. Blanton - ENTC Energy & Potential 22 Summary Energy conservation requires that Because of energy conservation the Coulomb force is conservative: Consequently,

Dr. Blanton - ENTC Energy & Potential 23 Stoke’s Theorem General mathematical theorem of Vector Analysis. any surface any vector field closed boundary of that surface a surface closed path C along boundary

Dr. Blanton - ENTC Energy & Potential 24 Since Stoke’s Theorem is of general validity, it can certainly be applied to the electrostatic fields: and

Dr. Blanton - ENTC Energy & Potential 25 any closed path through consequence of: energy conservation Coulomb’s force is conservative & fields are conservative

Dr. Blanton - ENTC Energy & Potential 26 Since C and S are arbitrary, this is only possible if:

Dr. Blanton - ENTC Energy & Potential 27 Recall that the curl of a gradient field is always zero. That is: Thus:

Dr. Blanton - ENTC Energy & Potential 28 The curl of the electrostatic field is zero. The circulation of any electrostatic field is zero An electrostatic field is non-rotational An electrostatic field is conservative. All of the preceding statements are equivalent.

Dr. Blanton - ENTC Energy & Potential 29 The complete set of postulates for electrostatics are: and These are Maxwell’s equations of Electrostatics The postulates are accepted to be true for any E or D field as long as  is independent of time.

Dr. Blanton - ENTC Energy & Potential 30 Cornerstones of Electrostatics Principle of Linear Superposition charges are the sources of the field Energy Conservation generalization for multiple charges SI units Coulomb’s Law conservative central force the field is non- rotational  Postulates of Electrostatics  theoretical experimental empirical facts field concept