In Ex. 3.2 we assumed that the
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- Problem 3.18 The potential at the surface of a sphere (radius R) is given by Vo = k cos 30, where k is a constant. Find the potential inside and outside the sphere, as well as the surface charge density o (0) on the sphere. (Assume there's no charge inside or outside the sphere.)arrow_forwardFind a potential V for the following force in spherical coordinates. 1 F = 2rsinée, +rcos Gê, + ê. r sineVerify that this force is conservative by explicitly showing that the curl is zero.arrow_forwardThis one is tougher! A sphere of radius r has charge q. (a) What is the infinitesimal increase in clectric potential energy dU if an infinitesimal amount of charge dq is brought to infinity to the surface of the sphere? (b) An uncharged sphere can acquire a total charge Q by the transfer of charge dq over and over and over. Use your answer to part a to find an cxpression for the potential energy of a uniformly-charged sphere of radius R with total charge Q. Answer: U = 3_1 Q² 5 4tc0 R' (c) Your answer to part b is the amount of energy nceded to assemble a charged sphere. It is often called the self-energy of the sphere. What is the self-energy of a proton, assuming it to be a charged sphere with a diamcter of 1.0 x 10 15 m?arrow_forward
- Problem 2.22 Find the potential a distances from an infinitely long straight wire that carries a uniform line charge λ. Compute the gradient of your potential, and check that it yields the correct field.arrow_forwardProblem 2.21 Find the potential inside and outside a uniformly charged solid sphere whose radius is R and whose total charge is q. Use infinity as your reference point. Compute the gradient of V in each region, and check that it yields the correct field. Sketch V(r).arrow_forwardProblem 2.20 One of these is an impossible electrostatic field. Which one? (a) E =k[xyÂ+2yzý+3xz2]; (b) E = k[y² + (2xy + z²)ý + 2yz 2). Here k is a constant with the appropriate units. For the possible one, find the potential, using the origin as your reference point. Check your answer by computing VV. [Hint: You must select a specific path to integrate along. It doesn't matter what path you choose, since the answer is path-independent, but you simply cannot integrate unless you have a particular path in mind.]arrow_forward
- Lets say there is a spherical thin shell of radius a. It carries uniform surface chargewith a density of ps C/m2, (p is ro). I want to find the potential V for points outside the spherical shell and inside the shell. (The reference point for V is set at infinity.) Plot V as a function of R. Thank you for helping me with this practice.arrow_forwardSubject: Ideal Conductors and Capacitors A ring of mass m and radius r has charge -Q uniformly distributed around it. The ring is located a distance h from an infinite grounded conducting plane. Let z be the vertical coordinate with z = 0 taken to be the center of the infinite conducting plane. Find the electric field above the conducting plane at points on the axis of the ring. Your answershould be a function of Q, r, and h.arrow_forwardSolve the shown integral and show is equal to one. prove it. please do then explain to me. thanks.arrow_forward
- Problem 3.36 (3rd edition): Two long straight wires, carrying opposite uniform line charges +1, are situated on either side of a long conducting cylinder (Fig. 3.39). The cylinder (which carries no net charge) has radius R, and the wires are a distance "a" from the axis. Find the potential at point 7. (Hint: you can use solution of problem 2.47) R a aarrow_forwardProblem 2.20 One of these is an impossible electrostatic field. Which one? (a) Ek[xy x + 2yzý + 3xz2]; (b) E= k[y² + (2xy + z²) ŷ + 2yz 2]. Here k is a constant with the appropriate units. For the possible one, find the potential, using the origin as your reference point. Check your answer by computing VV. [Hint: You must select a specific path to integrate along. It doesn't matter what path you choose, since the answer is path-independent, but you simply cannot integrate unless you have a particular path in mind.] Problem 2.11 Use Gauss's law to find the electric field inside and outside a spherical shell of radius R, which carries a uniform surface charge density o. Compare your answer to Prob. 2.7. Problem 2.21 Find the potential inside and outside a uniformly charged solid sphere whose radius is R and whose total charge is q. Use infinity as your reference point. Compute the gradient of V in each region, and check that it yields the correct field. Sketch V (r).arrow_forwardPA line of length L has a positive charge Q uniformly distributed over it. It is placed on the coordinate axes as shown. Note that the y axis bisects the line. Point P is placed a distance a 7. on the y axis from the line. Construct (but do not solve!) an integral to find the electric potential at point P. Include limits and express the integrals in terms of the given variables, constants, and the integration variable, x. х. +y a +x -L/2 L/2arrow_forward
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