Electric Potential of Charged Rod
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1 Electric Potential of Charged Rod Charge per unit length: λ = Q/L y dq = λ d Charge on slice d: dq = λd dv d L Electric potential generated by slice d: dv = kdq = kλd Electric potential generated by charged rod: V = kλ Z dl d d = kλ[ln ]dl d Limiting case of very short rod (L d): V = kλ ln = kλ [ln(d L) ln d] = kλln d L d 1 L «d kλ L d = kq d 2/9/2015 [tsl330 1/25]
2 Electric Potential of Charged Ring Total charge on ring: Q Charge per unit length: λ = Q/2πa Charge on arc: dq Find the electric potential at point P on the ais of the ring. dv = k dq r = Z V () = k kdq 2 a 2 dq 2 a 2 = Z k 2 a 2 dq = kq 2 a 2 2/9/2015 [tsl81 2/25]
3 Electric Potential of Charged Disk Area of ring: 2πada Charge on ring: dq = σ(2πada) Charge on disk: Q = σ(πr 2 ) Find the electric potential at point P on the ais of the disk. dv = k dq 2 a 2 = 2πσk ada 2 a 2 V () = 2πσk Z R 0 ada hp i R hp i 2 a = 2πσk 2 a 2 = 2πσk 2 R Electric potential at large distances from the disk ( R): V () = 2πσk 4 2s 3 1 R πσk»1 R = kσπr2 = kq 2/9/2015 [tsl82 3/25]
4 Electric Field and Electric Potential Determine the field or the potential from the source (charge distribution): E = 1 4πǫ 0 Z dq r 2 ˆr Determine the field from the potential: E = V î V y ĵ V z r de Determine the potential from the field: V = Systems with E = E ()î: E = dv d Application to charged ring: E = r dq Z r dv r 0 E d s Application to charged disk (at > 0):» E = 2πσk 1 2 R 2 V () = ˆk Z V = 1 4πǫ 0 0 E d kq ( 2 a 2 ) 3/2 V = kq 2 a 2 Z dq hp i V = 2πσk 2 R 2 r 2/9/2015 [tsl85 4/25]
5 Electric Potential and Electric Field in One Dimension (1) For given electric potential V ()find the electric field (a) E (1m), (b) E (3m). For given electric field E () and given reference potential potential V (0) = 0 find the electric potential (c) V (2m), (d) V (4m). 2 V[V] 2 E [V/m] [m] [m] 2/9/2015 [tsl88 5/25]
6 Electric Potential and Electric Field in One Dimension (2) For given electric potential V () find the electric field (a) E (0.5m), (c) E (2.5m), (b) E (1.5m), (d) E (3.5m). For given electric field E () and given reference potential potential V (0) = 0 find the electric potential (e) V (1m), (f) V (2m), (g) V (4m). 2 V[V] 2 E [V/m] [m] [m] 2/9/2015 [tsl89 6/25]
7 Electric Field from Electric Potential in Two Dimensions Given is the electric potential: V (, y) = a 2 by 3 with a = 1V/m 2, b = 1V/m 4. Find the electric field: E(, y) = E (, y)î E y (, y)ĵ via partial derivatives. E = V = 2a by3, E y = V y = 3by2 y V = 0 E = 1V/m E = 0 y V = 2V E = 3V/m E = 3V/m y V = 0 V = 1V E = 0 E = 2V/m E y= 0 E y= 0 0 1m 2/9/2015 [tsl86 7/25]
8 Electric Potential from Electric Field in Two Dimensions Given is the electric field: E = (2a by 3 )î 3by 2 ĵ with a = 1V/m 2, b = 1V/m 4. Find the electric potential V (, y) via integral along a specific path: Red path (0,0) (0, y) (, y): V (, y) = = 0 Z y 0 Z 0 E y (0, y)dy Z 0 E (, y)d (2a by 3 )d = a 2 by 3 (0,y) y (,y) Blue path (0,0) (,0) (, y): V (, y) = = Z 0 Z 0 E (,0)d (2a)d Z y 0 Z y 0 E y (, y)dy (3by 2 )dy = a 2 by 3 (0,0) (,0) 2/9/2015 [tsl87 8/25]
9 Electric Potential and Electric Field in Two Dimensions Given is the electric potential V (, y) = cy 2 with c = 1V/m 3. (a) Find the value (in SI units) of the electric potential V at point A. (b) Find the components E, E y (in SI units) of the electric field at point B. 2 y [m] A 1 B [m] 2/9/2015 [tsl91 9/25]
10 Electric Potential of a Charged Plane Sheet Consider an infinite plane sheet perpendicular to the -ais at = 0. The sheet is uniformly charged with charge per unit area σ. Electric field (magnitude): E = 2πk σ = σ 2ǫ 0 Direction: away from (toward) the sheet if σ > 0 (σ < 0). Electric field (-component): E = ±2πkσ. Electric potential: V = Z 0 E d = 2πkσ. Here we have used 0 = 0 as the reference coordinate. positively charged sheet E negatively charged sheet E V V 2/9/2015 [tsl92 10/25]
11 Electric Potential of a Uniformly Charged Spherical Shell Electric charge on shell: Q = σa = 4πσR 2 Electric field at r > R: E = kq r 2 Electric field at r < R: E = 0 Electric potential at r > R: V = Z r kq r 2 dr = kq r Electric potential at r < R: V = Z R kq r 2 dr Z r R (0)dr = kq R Here we have used r 0 = as the reference value of the radial coordinate. 2/9/2015 [tsl93 11/25]
12 Electric Potential of a Uniformly Charged Solid Sphere Electric charge on sphere: Q = ρv = 4π 3 ρr3 Electric field at r > R: E = kq r 2 Electric field at r < R: E = kq R 3 r Electric potential at r > R: V = Z r kq r 2 dr = kq r Electric potential at r < R: V = Z R Z kq r r 2 dr R kq R 3 r dr V = kq R kq `r2 2R 3 R 2 = kq 2R 3 r2 R 2 «2/9/2015 [tsl94 12/25]
13 Electric Potential of a Uniformly Charged Wire Consider a uniformly charged wire of infinite length. Charge per unit length on wire: λ (here assumed positive). Electric field at radius r: E = 2kλ r. Electric potential at radius r: V = 2kλ Z r r 0 1 r dr = 2kλ[ln r ln r 0] V = 2kλ ln r 0 r Here we have used a finite, nonzero reference radius r 0 0,. The illustration from the tetbook uses R ref for the reference radius, R for the integration variable, and R p for the radial position of the field point. 2/9/2015 [tsl95 13/25]
14 Electric Potential of Conducting Spheres (1) A conducting sphere of radius r 1 = 2m is surrounded by a concentric conducting spherical shell of radii r 2 = 4m and r 3 = 6m. The graph shows the electric field E(r). (a) Find the charges q 1, q 2, q 3 on the three conducting surfaces. (b) Find the values V 1, V 2, V 3 of the electric potential on the three conducting surfaces relative to a point at infinity. (c) Sketch the potential V (r). 5 E[V/m] r[m] 3 2/9/2015 [tsl96 14/25]
15 Electric Potential of Conducting Spheres (2) Consider a conducting sphere with radius r = 15cm and electric potential V = 200V relative to a point at infinity. (a) Find the charge Q and the surface charge density σ on the sphere. (b) Find the magnitude of the electric field E just outside the sphere. (c) What happens to the values of Q, V, σ, E when the radius of the sphere is doubled? 15cm 30cm 2/9/2015 [tsl97 15/25]
16 Electric Potential of Conducting Spheres (3) A spherical raindrop of 1mm diameter carries a charge of 30pC. (a) Find the electric potential of the drop relative to a point at infinity under the assumption that it is a conductor. (b) If two such drops of the same charge and diameter combine to form a single spherical drop, what is its electric potential? q1 V1 q V r r 1 q 2 V r 2 2/9/2015 [tsl98 16/25]
17 Electric Potential of Conducting Spheres (4) A positive charge is distributed over two conducting spheres 1 and 2 of unequal size and connected by a long thin wire. The system is at equilibrium. Which sphere (1 or 2)... (a) carries more charge on its surface? (b) has the higher surface charge density? (c) is at a higher electric potential? (d) has the stronger electric field net to it? 1 2 2/9/2015 [tsl99 17/25]
18 Unit Eam I: Problem #3 (Spring 11) Consider a region of space with a uniform electric field E = 0.5V/mî. Ignore gravity. (a) If the electric potential vanishes at point 0, what are the electric potentials at points 1 and 2? (b) If an electron (m = kg, q = C) is released from rest at point 0, toward which point will it start moving? (c) What will be the speed of the electron when it gets there? 5m y 4 E 3m m 2 1m 3m 5m 2/9/2015 [tsl403 18/25]
19 Unit Eam I: Problem #3 (Spring 11) Consider a region of space with a uniform electric field E = 0.5V/mî. Ignore gravity. (a) If the electric potential vanishes at point 0, what are the electric potentials at points 1 and 2? (b) If an electron (m = kg, q = C) is released from rest at point 0, toward which point will it start moving? (c) What will be the speed of the electron when it gets there? Solution: (a) V 1 = (0.5V/m)(2m) = 1V, V 2 = 0. 5m 3m y E 1m 2 1m 3m 5m 2/9/2015 [tsl403 18/25]
20 Unit Eam I: Problem #3 (Spring 11) Consider a region of space with a uniform electric field E = 0.5V/mî. Ignore gravity. (a) If the electric potential vanishes at point 0, what are the electric potentials at points 1 and 2? (b) If an electron (m = kg, q = C) is released from rest at point 0, toward which point will it start moving? (c) What will be the speed of the electron when it gets there? Solution: (a) V 1 = (0.5V/m)(2m) = 1V, V 2 = 0. (b) F = qe = qe î (toward point 3). 5m 3m 1m y m 3m 5m 1 E 2/9/2015 [tsl403 18/25]
21 Unit Eam I: Problem #3 (Spring 11) Consider a region of space with a uniform electric field E = 0.5V/mî. Ignore gravity. (a) If the electric potential vanishes at point 0, what are the electric potentials at points 1 and 2? (b) If an electron (m = kg, q = C) is released from rest at point 0, toward which point will it start moving? (c) What will be the speed of the electron when it gets there? Solution: (a) V 1 = (0.5V/m)(2m) = 1V, V 2 = 0. (b) F = qe = qe î (toward point 3). (c) V = (V 3 V 0 ) = 1V, U = q V = J, r 2K K = U = J, v = m = m/s. Alternatively: F = qe = N, a = F m = m/s 2, 5m 3m 1m y m 3m 5m 1 E = 2m, v = p 2a = m/s. 2/9/2015 [tsl403 18/25]
22 Unit Eam I: Problem #3 (Fall 10) An electron (m = kg, q = C) and a proton (m = kg, q = C) are released from rest midway between oppositely charged parallel plates. The plates are at the electric potentials shown. (a) Find the magnitude of the electric field between the plates. (b) What direction (left/right) does the electric field have? (c) Which particle (electron/proton/both) is accelerated to the left? (d) Why does the electron reach the plate before the proton? (e) Find the kinetic energy of the proton when it reaches the plate. 6V 0.2m 12V 2/9/2015 [tsl400 19/25]
23 Unit Eam I: Problem #3 (Fall 10) An electron (m = kg, q = C) and a proton (m = kg, q = C) are released from rest midway between oppositely charged parallel plates. The plates are at the electric potentials shown. (a) Find the magnitude of the electric field between the plates. (b) What direction (left/right) does the electric field have? (c) Which particle (electron/proton/both) is accelerated to the left? (d) Why does the electron reach the plate before the proton? (e) Find the kinetic energy of the proton when it reaches the plate. Solution: (a) E = 6V/0.2m = 30V/m. 6V 0.2m 12V 2/9/2015 [tsl400 19/25]
24 Unit Eam I: Problem #3 (Fall 10) An electron (m = kg, q = C) and a proton (m = kg, q = C) are released from rest midway between oppositely charged parallel plates. The plates are at the electric potentials shown. (a) Find the magnitude of the electric field between the plates. (b) What direction (left/right) does the electric field have? (c) Which particle (electron/proton/both) is accelerated to the left? (d) Why does the electron reach the plate before the proton? (e) Find the kinetic energy of the proton when it reaches the plate. Solution: (a) E = 6V/0.2m = 30V/m. (b) left 6V 0.2m 12V 2/9/2015 [tsl400 19/25]
25 Unit Eam I: Problem #3 (Fall 10) An electron (m = kg, q = C) and a proton (m = kg, q = C) are released from rest midway between oppositely charged parallel plates. The plates are at the electric potentials shown. (a) Find the magnitude of the electric field between the plates. (b) What direction (left/right) does the electric field have? (c) Which particle (electron/proton/both) is accelerated to the left? (d) Why does the electron reach the plate before the proton? (e) Find the kinetic energy of the proton when it reaches the plate. Solution: (a) E = 6V/0.2m = 30V/m. (b) left 6V 0.2m 12V (c) proton (positive charge) 2/9/2015 [tsl400 19/25]
26 Unit Eam I: Problem #3 (Fall 10) An electron (m = kg, q = C) and a proton (m = kg, q = C) are released from rest midway between oppositely charged parallel plates. The plates are at the electric potentials shown. (a) Find the magnitude of the electric field between the plates. (b) What direction (left/right) does the electric field have? (c) Which particle (electron/proton/both) is accelerated to the left? (d) Why does the electron reach the plate before the proton? (e) Find the kinetic energy of the proton when it reaches the plate. Solution: (a) E = 6V/0.2m = 30V/m. (b) left 6V 0.2m 12V (c) proton (positive charge) (d) smaller m, equal q larger q E/m 2/9/2015 [tsl400 19/25]
27 Unit Eam I: Problem #3 (Fall 10) An electron (m = kg, q = C) and a proton (m = kg, q = C) are released from rest midway between oppositely charged parallel plates. The plates are at the electric potentials shown. (a) Find the magnitude of the electric field between the plates. (b) What direction (left/right) does the electric field have? (c) Which particle (electron/proton/both) is accelerated to the left? (d) Why does the electron reach the plate before the proton? (e) Find the kinetic energy of the proton when it reaches the plate. Solution: (a) E = 6V/0.2m = 30V/m. (b) left 6V 0.2m 12V (c) proton (positive charge) (d) smaller m, equal q larger q E/m (e) K = q V = ( C)(3V) = J. 2/9/2015 [tsl400 19/25]
28 Intermediate Eam I: Problem #1 (Spring 06) Consider a point charge q = 8nC at position = 4m, y = 0 as shown. (a) Find the electric field components E and E y at point P 1. (b) Find the electric field components E and E y at point P 2. (c) Find the electric potential V at point P 3. (d) Find the electric potential V at point P 2. P 2 y 4m P 1 3m P 3 8nC 2/9/2015 [tsl348 20/25]
29 Intermediate Eam I: Problem #1 (Spring 06) Consider a point charge q = 8nC at position = 4m, y = 0 as shown. (a) Find the electric field components E and E y at point P 1. (b) Find the electric field components E and E y at point P 2. (c) Find the electric potential V at point P 3. (d) Find the electric potential V at point P 2. P 2 y 4m P 1 Solution: (a) E = 0, E y = k 8nC (3m) 2 = 7.99N/C. 3m P 3 8nC 2/9/2015 [tsl348 20/25]
30 Intermediate Eam I: Problem #1 (Spring 06) Consider a point charge q = 8nC at position = 4m, y = 0 as shown. (a) Find the electric field components E and E y at point P 1. (b) Find the electric field components E and E y at point P 2. (c) Find the electric potential V at point P 3. (d) Find the electric potential V at point P 2. P 2 y 4m P 1 Solution: (a) E = 0, E y = k 8nC (3m) 2 = 7.99N/C. 3m (b) E = k 8nC (5m) 2 cos θ = 2.88N/C 4 5 = 2.30N/C. E y = k 8nC (5m) 2 sin θ = 2.88N/C 3 5 = 1.73N/C. P 3 8nC 2/9/2015 [tsl348 20/25]
31 Intermediate Eam I: Problem #1 (Spring 06) Consider a point charge q = 8nC at position = 4m, y = 0 as shown. (a) Find the electric field components E and E y at point P 1. (b) Find the electric field components E and E y at point P 2. (c) Find the electric potential V at point P 3. (d) Find the electric potential V at point P 2. P 2 y 4m P 1 Solution: (a) E = 0, E y = k 8nC (3m) 2 = 7.99N/C. 3m (b) E = k 8nC (5m) 2 cos θ = 2.88N/C 4 5 = 2.30N/C. E y = k 8nC (5m) 2 sin θ = 2.88N/C 3 5 = 1.73N/C. P 3 8nC (c) V = k 8nC 4m = 17.98V. 2/9/2015 [tsl348 20/25]
32 Intermediate Eam I: Problem #1 (Spring 06) Consider a point charge q = 8nC at position = 4m, y = 0 as shown. (a) Find the electric field components E and E y at point P 1. (b) Find the electric field components E and E y at point P 2. (c) Find the electric potential V at point P 3. (d) Find the electric potential V at point P 2. P 2 y 4m P 1 Solution: (a) E = 0, E y = k 8nC (3m) 2 = 7.99N/C. 3m (b) E = k 8nC (5m) 2 cos θ = 2.88N/C 4 5 = 2.30N/C. E y = k 8nC (5m) 2 sin θ = 2.88N/C 3 5 = 1.73N/C. P 3 8nC (c) V = k 8nC 4m = 17.98V. (d) V = k 8nC 5m = 14.38V. 2/9/2015 [tsl348 20/25]
33 Unit Eam I: Problem #1 (Spring 09) Consider two point charges positioned on the -ais as shown. (a) Find magnitude and direction of the electric field at point P. (b) Find the electric potential at point P. (c) Find the electric potential energy of an electron (mass m = kg, charge q = C) when placed at point P. (d) Find magnitude and direction of the acceleration the electron eperiences when released at point P. 8nC 8nC P 2m 2m 2/9/2015 [tsl389 21/25]
34 Unit Eam I: Problem #1 (Spring 09) Consider two point charges positioned on the -ais as shown. (a) Find magnitude and direction of the electric field at point P. (b) Find the electric potential at point P. (c) Find the electric potential energy of an electron (mass m = kg, charge q = C) when placed at point P. (d) Find magnitude and direction of the acceleration the electron eperiences when released at point P. Solution: 8nC 8nC P (a) E = k 8nC (4m) 2 k(8nc) = 4.5N/C 18N/C = 13.5N/C (directed left). (2m) 2 2m 2m 2/9/2015 [tsl389 21/25]
35 Unit Eam I: Problem #1 (Spring 09) Consider two point charges positioned on the -ais as shown. (a) Find magnitude and direction of the electric field at point P. (b) Find the electric potential at point P. (c) Find the electric potential energy of an electron (mass m = kg, charge q = C) when placed at point P. (d) Find magnitude and direction of the acceleration the electron eperiences when released at point P. Solution: 8nC 8nC P (a) E = k 8nC (4m) 2 k(8nc) = 4.5N/C 18N/C = 13.5N/C (directed left). (2m) 2 (b) V = k 8nC 4m k(8nc) 2m = 18V 36V = 18V. 2m 2m 2/9/2015 [tsl389 21/25]
36 Unit Eam I: Problem #1 (Spring 09) Consider two point charges positioned on the -ais as shown. (a) Find magnitude and direction of the electric field at point P. (b) Find the electric potential at point P. (c) Find the electric potential energy of an electron (mass m = kg, charge q = C) when placed at point P. (d) Find magnitude and direction of the acceleration the electron eperiences when released at point P. Solution: 8nC 8nC P (a) E = k 8nC (4m) 2 k(8nc) = 4.5N/C 18N/C = 13.5N/C (directed left). (2m) 2 (b) V = k 8nC 4m k(8nc) 2m = 18V 36V = 18V. (c) U = qv = (18V)( C) = J. 2m 2m 2/9/2015 [tsl389 21/25]
37 Unit Eam I: Problem #1 (Spring 09) Consider two point charges positioned on the -ais as shown. (a) Find magnitude and direction of the electric field at point P. (b) Find the electric potential at point P. (c) Find the electric potential energy of an electron (mass m = kg, charge q = C) when placed at point P. (d) Find magnitude and direction of the acceleration the electron eperiences when released at point P. Solution: 8nC 8nC P (a) E = k 8nC (4m) 2 k(8nc) = 4.5N/C 18N/C = 13.5N/C (directed left). (2m) 2 (b) V = k 8nC 4m k(8nc) 2m = 18V 36V = 18V. (c) U = qv = (18V)( C) = J. (d) a = qe m = ( C)(13.5N/C) kg 2m 2m = ms 2 (directed right). 2/9/2015 [tsl389 21/25]
38 Electric Dipole Field L E E q 0 q p E =» kq ( L/2) 2 kq ( L/2) 2 ( L/2) 2 = kq ( L/2) 2 ( L/2) 2 ( L/2) 2 2kqL 3 = 2kp 3 (for L) = 2kqL ( 2 L 2 /4) 2 Electric dipole moment: p = q L Note the more rapid decay of the electric field with distance from an electric dipole ( r 3 ) than from an electric point charge ( r 2 ). The dipolar field is not radial. 2/9/2015 [tsl23 22/25]
39 Electric Dipole Potential Use spherical coordinates: V = V (r, θ) independent of azimuthal coordinate φ. q Superposition principle: V = V V = k (q) «= kq r r r r r r Large distances (r L): r r L cos θ, r r r 2 V (r, θ) k Electric dipole moment: p = ql (magnitude) Electric dipole potential: V (r, θ) k pcos θ r 2 ql cos θ r 2 q 0 L θ r r q r 2/9/2015 [tsl100 23/25]
40 Electric Potential Energy of Two Point Charges Consider two different perspectives: #1a Electric potential when q 1 is placed: V ( r 2 ). = V 2 = k q 1 r 12 Electric potential energy when q 2 is placed into potential V 2 : U = q 2 V 2 = k q 1q 2 r 12 #1b Electric potential when q 2 is placed: V ( r 1 ). = V 1 = k q 2 r 12 Electric potential energy when q 1 is placed into potential V 1 : U = q 1 V 1 = k q 1q 2 r 12. #2 Electric potential energy of q 1 and q 2 : z q 1 U = 1 2 2X q i V i, i=1 where V 1 = k q 2 r 12, V 2 = k q 1 r 12. r 1 r 2 r 12 q 2 y 2/9/2015 [tsl101 24/25]
41 Electric Potential Energy of Three Point Charges #1 Place q 1, then q 2, then q 3, and add all changes in potential energy: U = 0 k q 1q 2 q1 q 3 k q «2q 3 q1 q 2 = k q 1q 3 q «2q 3. r 12 r 13 r 23 r 12 r 13 r 23 #2 Symmetric epression of potential energy U in terms of the potentials V i eperienced by point charges q 1 : U = 1 3X q1 q 2 q i V i = k q 1q 3 q «2q 3, 2 r 12 r 13 r 23 where V 1 = k V 2 = k V 3 = k i=1 q2 q «3, r 12 r 13 q1 q «3, r 12 r 23 q1 q «2. r 13 r 23 r 13 z q 1 q 3 r 23 r 12 q 2 y 2/9/2015 [tsl102 25/25]
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