UNIVERSITY PHYSICS II CHAPTER 17 Electrical Potential Energy nd Electic pote s 17.1 Electrical potential energy and electric potential 1. Electric force is conservative force Coulomb’ s force:F= L Source charge: E Test charge: q OLF=qE araH=F.d=nEd≈9QF·d_gQdr W=l dn g @dr 92 1 4丌Enr24m。r
1 1. Electric force is conservative force ) 1 1 ( 4 4 d d 4 d 4 ˆ d d d d 0 2 0 2 0 2 0 i f L r r r r qQ r qQ r W W r qQ r r qQr r W F r qE r f i = = = − = ⋅ = ⋅ = ⋅ = ∫ ∫ πε πε πε πε r r r r r Source charge： Test charge： q Q F qE r r = ' Q r r r a r b a r r r r b r r d q E r L dr + + §17.1 Electrical potential energy and electric potential Coulomb’s force: 2 21 0 21 ˆ 4 1 r r qQ F πε = r
al potential energy and lectric potential 2. Electrical potential energy +conserv=noncon +(-4PE)=4KE ∴W W=-(PE-PE=-APE Edr 4兀Eor; (PE-PE) 4 The electrical potential f charge q PE qe 1 4 s 17.1 Electrical potential energy and electric potential 3. The electric potential and electric potent difference Divided the equation by the charge q PE PE E·dr= 478 q Define PE The electric potential v q The electric potential v at a point in an ric field is the potential per unit charge at that point the potential energy of q PEoa=qI
2 ) ( ) 1 1 ( 4 ) 1 1 ( 4 d 0 0 f i f i f i i f PE PE r r qQ r r qQ W q E r = − − = − − = ⋅ = − ∫ πε πε r r ∴ Wconserv = W = −(PEf − PEi) = −∆PE Q Wnoncon +Wconserv = Wnoncon + (−∆PE) = ∆KE 2. Electrical potential energy The electrical potential energy of charge q r qQ PE 1 4πε 0 = §17.1 Electrical potential energy and electric potential Divided the equation by the charge q ) ( ) 1 1 ( 4 d 0 f i q PE q PE r r Q E r i f f i ⋅ = − − = − − ∫ πε r r 3. The electric potential and electric potential difference The electric potential V at a point in an electric field is the potential energy per unit charge at that point. The electric potential q PE V of q = Define: the potential energy of q PEof q = qV §17.1 Electrical potential energy and electric potential
817. 1 Electrical potential energy and electric potential Define the electric potential difference ∫E.d=vr-H1)=-( PE PE V -V=lE.dr OThe electric potential is not same thing as the electric potential energy Electric potential is a property of a point in space, whether or not a charge is placed at that point. It can be positive, negative or zero 817.1 Electrical potential energy and electric potential @In PE=gV, pe depends on both the sign of q and the sign of v at the point where q is placed; @the electric potential and the electric potential energy are scalar quantities. @The place where the electric potential and the electric potential energy is set to zero is arbitrary, we can choose it anywhere we like for a finite charge distribution The unit of the electric potential: volt(v=J/C
3 Define the electric potential difference d ( ) ( ) f i q PE q PE E r V V i f ⋅ = − f − i = − − ∫ r r ∫ − = ⋅ f i V V E dr i f r r or Note: 1The electric potential is not same thing as the electric potential energy.Electric potential is a property of a point in space, whether or not a charge is placed at that point. It can be positive, negative or zero. §17.1 Electrical potential energy and electric potential 2In PE=qV, PE depends on both the sign of q and the sign of V at the point where q is placed; 3the electric potential and the electric potential energy are scalar quantities. 4The place where the electric potential and the electric potential energy is set to zero is arbitrary, we can choose it anywhere we like for a finite charge distribution. The unit of the electric potential: volt(v)=J/C §17.1 Electrical potential energy and electric potential
817. 1 Electrical potential energy and electric potential XThe first method of calculating the electric potential V-V=「E·dF Path Field line dr 817.2 The calculation of the electric potential 1. The electric potential of a pointlike charge q Edr rdr= )=V-V teo r
4 ∫ − = ⋅ f i V V E dr i f r r r r d ※The first method of calculating the electric potential §17.1 Electrical potential energy and electric potential §17.2 The calculation of the electric potential 1. The electric potential of a pointlike charge Q P r ir q E r r r d i f i f r r V V r r Q r r r Q E r f i ⋅ = ⋅ = − = − ∫ ∫ ) 1 1 ( 4 ˆ d 4 d 0 2 0 f i πε πε r r r
817.2 The calculation of the electric potential Choose voa=K(o=0, the electric potential of a point charge o is V(r=. or v(r) 0 817.2 The calculation of the electric potential Note (i)It is obviously that when r-yo0, v approaches 0. (iithe electric potential varies inversely the distance r to the first power, and the electric field varies inversely the distance r to the second power (ii) the electric potential is scalar, the electric field is vector
5 Choose Vf =V(rf )=V(∞)=0, the electric potential of a point charge Q is r Q V r r Q V r i i 0 0 4 ( ) 4 ( ) πε πε = or = + Q x y V(r) − Q §17.2 The calculation of the electric potential (i)It is obviously that when r→∞, V approaches 0. (ii)the electric potential varies inversely the distance r to the first power, and the electric field varies inversely the distance r to the second power. (iii) the electric potential is scalar, the electric field is vector. Note: §17.2 The calculation of the electric potential
817.2 The calculation of the electric potential 2. The electric potential of a collection of pointlike charges The total electric potential is the algebraic scalar sum of the potential s of each charges V=V1+V2+V3 十 十 十 4兀Enr14znr,4 3. The electric potential of continuous charge distributions of finite size do dv= V= itor 4Te. Distr.r 817.2 The calculation of the electric potential XXThe second method of calculating the electric potential do do d=-p= 4 兀Ear 47e distr. r Note This method can be applied only to the finite charge distribution. If the charge distribution is infinite, we must use the integral of the electric field as in the definition of the electric potentiaL. lel:P77317.2 Example 2: P773173 6
6 2. The electric potential of a collection of pointlike charges The total electric potential is the algebraic scalar sum of the potential s of each charges. L L = + + + = + + + 0 3 3 0 2 2 0 1 1 1 2 3 4 4 4 r Q r Q r Q V V V V πε πε πε 3. The electric potential of continuous charge distributions of finite size ∫ = = distr. charge 0 0 d 4 1 4 d d r Q V r Q V πε πε §17.2 The calculation of the electric potential Note: This method can be applied only to the finite charge distribution. If the charge distribution is infinite, we must use the integral of the electric field as in the definition of the electric potential. Example 1: P773 17.2 Example 2: P773 17.3 ※The second method of calculating the electric potential ∫ = = distr. charge 0 0 d 4 1 4 d d r Q V r Q V πε πε §17.2 The calculation of the electric potential
817.2 The calculation of the electric potential @ Find the electric potential a distance r from a spherical shell of radius r that has a charge o distributed uniformly throughout its surface for r>R and rR) 4 E choose V=0 E CC edr Qr·dr oc .7"y nE 0 R 兀G 817.2 The calculation of the electric potential outside 47er R E E·dr inside E =「E,drF+「Ed R Qr·drQ 4760/ 4ER 4rE,R constant O R
7 1Find the electric potential a distance r from a spherical shell of radius R that has a charge Q distributed uniformly throughout its surface, for r >R and r < = πε r R Q o r P r E r o 2 1 r ∝ r E R = 0 V∞ choose §17.2 The calculation of the electric potential constant 4 4 ˆ d d d d 0 2 0 in out inside ' ' = = ⋅ = = ⋅ + ⋅ = ⋅ ∫ ∫ ∫ ∫ ∞ ∞ ∞ R Q r Qr r E r E r V E r R R R P P πε πε r r r r r r r r r Q V 1 4 0 outside = ∝ πε R Q o r P r E r o 2 1 r ∝ r E R P′ r ∝ 1 o R r R Q 0 4πε V §17.2 The calculation of the electric potential
817.2 The calculation of the electric potential Q Find the electric potential a distance r from the center of a sphere of radius r that has charge o distributed uniformly throughout its volume, forr >R and r<R, respectively r<R E=arE R R Vowie-jEu dr-J Are y? choose v =0 CC- 4 817.2 The calculation of the electric potential E·dr Q R En·dF+「EdF RQr:d,rQ·dr 4 0 R 4兀Enr If we choose R V=0.v inside 十 4E。R3224zEnR 4E。2R 3-02) As shown in Fig. 17.20 8
8 2Find the electric potential a distance r from the center of a sphere of radius R that has a charge Q distributed uniformly throughout its volume, for r >R and r < = r R r Qr r r R R Q E πε πε r r r r Q r Qr r V E r P r 1 4 4 ˆ d d 0 2 0 outside out = ∝ ⋅ = ⋅ = ∫ ∫ ∞ ∞ πε πε r r r = 0 V∞ choose Q R O r P §17.2 The calculation of the electric potential Q R O r P (3 ) 4 2 1 4 ) 2 2 ( 4 4 ˆ d 4 d d d d 2 2 0 0 2 2 3 0 2 0 3 0 in out inside ' ' R r R Q R R r Q R Q r Qr r R Qr r E r E r V E r R R r R R P P = − = − + ⋅ + ⋅ = = ⋅ + ⋅ = ⋅ ∫ ∫ ∫ ∫ ∫ ∞ ∞ ∞ πε πε πε πε πε r r r r r r r r r As shown in Fig.17.20 §17.2 The calculation of the electric potential If we choose VO=0,Vinside=?
817.2 The calculation of the electric potential 3 Calculate the electric potential at a point x between two infinite, uniformly charged plates, separated by a distance d 十 E (0 region xd =∫E=+E)d2= 5050 region 0<x<d V=E·dF=(-Ei),di y=—x Eo (d)-V(0)=d=Ed
9 ⎪ ⎩ ⎪ ⎨ ⎧ − d V E r x Ei xi x d d d d x 0 0 0 f 0 i d ˆ ) d ˆ d 0d ( ε σ ε σ = ⋅ = + − ⋅ = − = ∫ ∫ ∫ ∫ r r region 0 < x < d x x V E r Ei xi x x 0 0 0 f 0 i d ˆ ) d ˆ d ( ε σ ε σ = − = = ⋅ = − ⋅ ∫ ∫ ∫ r r ⋅ − σ O x + σ x V O d V d −V = d = Ed 0 ( ) (0) ε σ §17.2 The calculation of the electric potential
817.2 The calculation of the electric potential @the electric potential due to an electric dipole ∑V=V++ 十 4Ta r-)- 4 817.2 The calculation of the electric potential If r>>d -r、≈dcos6 ≈r d cos e 4 I Pcos 6 4T8 10
10 4the electric potential due to an electric dipole ( ) ( ) ( ) ( ) 0 0 ( ) ( ) ( ) ( ) 2 1 4 ( ) 4 1 + − − + + − + − = − = − = + = ∑ = + r r q r r r q r q V V V V i i πε πε §17.2 The calculation of the electric potential If r >> d 2 0 2 0 2 ( ) ( ) ( ) ( ) cos 4 1 cos 4 cos r p r q d V r r r r r d θ πε θ πε θ = = ≈ − ≈ − + − + §17.2 The calculation of the electric potential