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24100 Electricity & Op-cs Lecture 6 – Chapter 23 sec. 1-3

Fall 2017 Semester Professor KolAck done by a

• Definitions from mechanics (Physics 152/172):

How much work does /ℓ poor Sisyphus do? 1% =−23 ∙1ℓ What is the change in the potential of the rock? . 1* =−23 ∙1ℓ

• : Net work independent of the path.

– Total work done: %&'( = *+ −*- (change in ) Work Done by Electrostatic Force

• An electric exerts a force on a charge.

8>9 /ℓ 23 7

• If we push it to the left, we do work on the charge: 1% =−23 ∙1ℓ>0 – The charge gains potential energy. • If we release the charge, it accelerates to the right – The charge loses potential energy. – It gains (total energy is conserved). ! : change in kinetic energy of a particle = - change in potential energy of a particle ! change in kinetic energy of a particle = work done by E- field

Potential Energy vs

• Electric potential difference: SI units: per

∆; = ? 1; =−? 7 ∙1ℓ= ; −;! ! ! – This only depends on the electric potential at points < and =… – Electric potential is a property of the field. • Difference in potential energy: ∆* = : ∆; SI units: Joules – This depends on the charge, :… – It is not a property of the field. SI Units

• Units of electric potential: Joules per Coulomb

• We will use this so frequently that we define: 1 = 1 /Coulomb • : 1 Newton/Coulomb = 1 Volt/meter Electric Potential Energy

! Electrostatic force is conservative force

• circulation of electrostatic field is zero

E dl =0 · Z NB: there is inductive E-field, for which work over closed path is not 0 (circulation not 0)

2

Lecture 6-8 Electric Potential Energy and Electric Potential High U (potential energy) positive Low U charge q0 High V (potential) High V negative

charge q0 Low U

Low V High U Low V

Electric field direction Electric field direction qq Ur() k 0 r

Urq () q Vr()0 k qr0 Properties of the Electric Potential

• The electric field points in the direction of decreasing electric potential. – As Sisyphus’ rock rolls down hill (direction of 23) it loses potential energy. • So far, our definition only referred to changes in potential energy and differences in electric potential. – You can add an arbitrary constant to the electric potential without changing the potential difference. – But it must be the same value at all points in space. • We usually define the electric potential as the potential difference relative to a convenient “reference point”. Calculating Electric Potential

• If the electric potential is a property of the field, how do we calculate it? • The electric potential of a field at a point "3 is the work per unit charge required to move from the reference point, "3#'+, to the point, "3: % $3 =; "3 −; =−? 7 ∙1ℓ : #'+ $3%&'

• Maybe it would be nice to pick "3#'+ so that ;#'+ =0. Lecture 6-12 Potential at P due to a point charge q From ∞

Ur() Vr()= q0 q0 q = k r Electric Potential due to a Point Charge 1 ! , 7 (= $ (̂ + 4/01 ( 1ℓ =−1( (̂ ( (#'+

# ; (−;(#'+ =−? 7 (∙1ℓ #%&' 3 # 9# 3 " " =− 8 : = − 4567 #%&' # 4567 # #%&'

• We can make ; (#'+ = 0 if we let (#'+ → ∞. Electric Field from a Point Charge

• Electric potential: 1 ! ; (= 4/01 ( • Electric field: [; 1 ! 7 =−Z;=− Z(= $ (̂ [( 4/01 ( V(r) versus r for a positive charge at r = 0

V(r) to kq r

r For a point charge V(r 0)

1/31/2016 22 Demos: + + + R + + + + + R + + + + + + + + + V(r) + kQ + kQ V(R) + R R

Gauss’ law says the sphere looks like a point charge outside R.

1/31/2016 24

Lecture 6-13 Volt • V=U/q is measured in => 1 V (volt) = 1 J / 1 C JNm⋅ [VEmV] == =[ ]⋅ = POTENTIALCC DIFFERENCES V2 – V1 NV [E] == Cm 111JCV= ⋅ 1eV≡⋅≅ | e | 1 V 1.602 × 10−19 C ⋅ 1 V (electron volt)

• V depends on an arbitrary choice of the reference point. • V is independent of a test charge with which to measure it. Example This enclosure has an electric The walls of this room have an potential of -750 kV. electric potential of zero volts.

Negatively charged H- hydrogen are produced in the enclosure.

How much kinetic energy do they have when the leave the room? Lecture 6-1 Electric Potential Energy of a Charge in Electric Field

• Coulomb force is conservative => Work done by the Coulomb

force is path independent. dl

• an associate potential energy Ur () to charge q0 at any point r in space.

dW q0 E d l Work done by E field Potential energy change dU dW q0 E dl of the charge q0

It’s energy! A measured in J (Joules) Lecture 6-2 Electric Potential Energy of a Charge (continued)

UUrUi() () 0 r qEdl0 i

i is “the” reference point. dl Choice of reference point (or point of zero potential energy) is arbitrary.

dW q0 E d l i is often chosen to be dU dW q E dl infinitely far ( ) 0 Lecture 6-12 Calculating the Field from the Potential (1)

• We can calculate the electric field from the electric potential starting with W V e, dW qE ds q • Which allows us to write

qdV qE ds E ds dV

• If we look at the component of the electric field along the direction of ds, we can write the magnitude of the electric field as the partial derivative along the direction s V E S s Lecture 6-16 E from V

We can obtain the electric field E from the potential V by inverting the integral that computes V from E:

→ → r → r V (r) = − E ⋅d l = − (E dx + E dy + E dz) ∫ ∫ x y z ∂V ∂V ∂V E = − Ey = − Ez = − x ∂x ∂y ∂z Expressed as a vector, E is the negative gradient of V ! ! E = −∇V Electric Field (again) • We can calculate the electric field from the electric potential: [; [; [; 7 =−Z ; = − \̂ + ^̂ + `a [" [< [_ • This is called the “gradient” of the electric potential. • Useful relations: [;(() [; [( = The “chain rule”… [" [( [" Also works for < and _. b# $ b# c b# d (= "$ +<$ +_$ = = = b$ # bc # bd # Z( = (̂ Constant Electric Field

• What electric potential produces a constant electric field, 7 =7\?̂ • Electric field: 7 =−Z; be be be – What function will have =−7, =0, =0? b$ bc bd – How about ; "3 =−7"... • A linear electric potential produces a constant electric field. Electric Potential due to several Point Charges • Electric potential for a single point charge: 1 ! ; (= (when (#'+ →∞) 4/01 ( • Electric potential at point "3 due to several point charges: 1 ! ; "3 = ; - 4/01 (- - where (- = "3 −"3- is the distance to each charge. • ; "3 is a scalar function (no direction). Electrical Potential Energy

Push q0 “uphill” and its electrical potential energy increases according to

kq q U 0 r

The work required to move q0 initially at rest at is kq q W 0 . r Work per unit charge is kq V . r

1/31/2016 23 Lecture 6-9 Potential Energy of a Multiple-Charge Configuration

(a) kq12 q/ d

(b) qq qq q q kk1 2 + 13+ k 2 3 d d 2d

qq12qq13 qq 24 qq 34 (c) kkkk+ + ddd d qq qq +kk23+ 14 22dd Example: Electric Potential Energy What is the change in electrical potential energy of a released electron in the atmosphere when the electrostatic force from the near ’s electric field (directed downward) causes the electron to move vertically upwards through a distance d? 1. DU of the electron is related to the work done on it by the electric field:

2. Work done by a constant force on a particle undergoing displacement:

3. Electrostatic Force and Electric Field are related:

1/31/2016 11 Demo DU q DV Get energy out

charge flow kQ V + R + + R + + + r r1 2 + + DV V(r ) V(r ) + + 1 2 + across fluorescent light bulb Also try an elongated neon bulb.

1/31/2016 25 Lecture 6-13 Math Reminder - Partial Derivatives

• Given a function V(x,y,z), the partial derivatives are

V V V they act on x, y, and z independently

x y x

• Example: V(x,y,z) = 2xy2 + z3 V 2 2y Meaning: partial derivatives x give the slope along the V 4xy respective direction y V 3z 2 z Lecture 6-14 Calculating the Field from the Potential (2)

• We can calculate any component of the electric field by taking the partial derivative of the potential along the direction of that component • We can write the components of the electric field in terms of partial derivatives of the potential as

VVV EEExyz; ; xyz • In terms of graphical representations of the electric potential, we can get an approximate value for the electric field by measuring the gradient of the potential perpendicular to an line

EV also written as EV