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 r< respectively 0 (r<R) E Q (r>R) 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 <R respectively. r r Q r Qr r V E r P r 1 4 4 ˆ d d 0 2 0 out out = ∝ ⋅ = ⋅ = ∫ ∫ ∞ ∞ πε πε r r r ( ) 4 ˆ 0 ( ) 2 0 r R r Qr r R E > < = 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, respectively. ( ) 4 ˆ ( ) 4 2 0 3 0 ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ > < = 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<x< 与 (x<0,x> region x< 0 =「E=[odx=0 V=0 817.2 The calculation of the electric potential region r>d =∫E=+E)d2= 5050 region 0<x<d V=E·dF=(-Ei),di y=—x Eo (d)-V(0)=d=Ed
9 ⎪ ⎩ ⎪ ⎨ ⎧ < > − < < = 0 ( 0, ) (0 ) 0 x x d x d E ε σ §17.2 The calculation of the electric potential 3Calculate the electric potential at a point x between two infinite, uniformly charged plates, separated by a distance d. ⋅ + σ O x − σ region x < 0 d 0d 0 0 0 = ⋅ = = ∫x ∫x V E r x r r VO = 0 region x > 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