Dr Hugh Blanton ENTC 3331 Energy Potential The

Dr. Hugh Blanton ENTC 3331

Energy & Potential

• The work done, or energy expended, in moving any object a vector differential distance dl under the influence of a force is: • Work, or energy, is measured in joules (J). Dr. Blanton - ENTC 3331 - Energy & Potential 3

• The differential electric potential energy d. W per unit charge is called the differential electric potential (or differential voltage) d. V. minus sign is convention • Convention: potential at infinity 0 ground. Dr. Blanton - ENTC 3331 - Energy & Potential 4

• Electrical potential of a point charge & E-field of a point charge, q, located at the origin? Dr. Blanton - ENTC 3331 - Energy & Potential 5

• Compare the electric field and potential of a point charge. • Expressions are rather similar. Dr. Blanton - ENTC 3331 - Energy & Potential 6

• Refer to the gradient in the spherical coordinates no dependence on q or f Dr. Blanton - ENTC 3331 - Energy & Potential 7

• It follows that for a point charge: Dr. Blanton - ENTC 3331 - Energy & Potential 8

• Generalizations: • All complicated charge distributions are always superposition of fields due to many point charges. • All these fields superimpose linearly. Dr. Blanton - ENTC 3331 - Energy & Potential 9

• Multiple charges • Charge distributions Dr. Blanton - ENTC 3331 - Energy & Potential 10

• And importantly, • for all distributions. • Note {do not confuse with potential energy} Coulomb force Dr. Blanton - ENTC 3331 - Energy & Potential potential energy 11

• From mechanics Gravitational force potential energy • For both Coulomb and mechanical forces, the force is equal to the gradient of the potential energy. Dr. Blanton - ENTC 3331 - Energy & Potential 12

• This isn’t so surprising, since • Coulomb’s law • And the gravitational force • are so similar. Dr. Blanton - ENTC 3331 - Energy & Potential 13

• Forces for which conservative. holds are • That is, the energy associated with the conservative force is conserved. Dr. Blanton - ENTC 3331 - Energy & Potential 14

• 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 3331 - Energy & Potential 15

• Important observation • while Dr. Blanton - ENTC 3331 - Energy & Potential 16

• Find the E-field above a circular disk of radius a with r. S = constant. • What are the potential, V, and E-field at point P(0, 0, z) right above the circular disk? z y x Dr. Blanton - ENTC 3331 - Energy & Potential 17

z y x Dr. Blanton - ENTC 3331 - Energy & Potential 18

• The question is, what is the field Dr. Blanton - ENTC 3331 - Energy & Potential 19

• Since Dr. Blanton - ENTC 3331 - Energy & Potential 20

• 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 3331 - Energy & Potential 21

Summary • Energy conservation requires that • Because of energy conservation the Coulomb force is conservative: • Consequently, Dr. Blanton - ENTC 3331 - Energy & Potential 22

Stoke’s Theorem • General mathematical theorem of Vector Analysis. any vector field any surface closed boundary of that surface closed path C along boundary a surface Dr. Blanton - ENTC 3331 - Energy & Potential 23

• Since Stoke’s Theorem is of general validity, it can certainly be applied to the electrostatic fields: • • and Dr. Blanton - ENTC 3331 - Energy & Potential 24

any closed path through consequence of: • energy conservation • Coulomb’s force is conservative & fields are conservative Dr. Blanton - ENTC 3331 - Energy & Potential 25

• Since C and S are arbitrary, this is only possible if: Dr. Blanton - ENTC 3331 - Energy & Potential 26

• Recall that the curl of a gradient field is always zero. • That is: • Thus: Dr. Blanton - ENTC 3331 - Energy & Potential 27

• 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 3331 - Energy & Potential 28

• 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 r is independent of time. Dr. Blanton - ENTC 3331 - Energy & Potential 29

Cornerstones of Electrostatics ge ive mu lt ne liz at e c ion ha rg es ipl t va ra SI units e r rc se o f n l co ra t n ce empirical facts Energy Conservation experimental Principle of Linear Superposition for Postulates of Electrostatics Dr. Blanton - ENTC 3331 - Energy & Potential the field is nonrotational 30 field concept charges are the sources of the field theoretical Coulomb’s Law