3.8.1 Electric Potential Due to a System of Two Charges. Figure Electric dipole

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Charity Ross
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1 3.8 Solved Poblems Electic Potential Due to a System o Two Chages Conside a system o two chages shown in Figue Figue Electic dipole Find the electic potential at an abitay point on the x axis and make a plot. Solution: The electic potential can be ound by the supeposition pinciple. At a point on the x axis, we have 1 q 1 ( q) q 1 1 V( x) = + = 4 πε x a 4 πε x+ a 4 πε x a x+ a The above expession may be ewitten as V( x) 1 1 = V x/ a 1 x/ a+ 1 whee V = q/4 πε a. The plot o the dimensionless electic potential as a unction o x/a. is depicted in Figue Figue

2 As can be seen om the gaph, V( x ) diveges at x/ a= ± 1, whee the chages ae located Electic Dipole Potential Conside an electic dipole along the y-axis, as shown in the Figue Find the electic potential V at a point P in the x-y plane, and use V to deive the coesponding electic ield. By supeposition pinciple, the potential at P is given by Figue V 1 q q = Vi = i 4 πε + whee ± = +. I we take the limit whee a, then a acosθ 1 1 1/ 1 1 = 1 ( a/ ) ( a/ )cosθ 1 ( a/ ) ( a/ )cosθ + = ± + ± and the dipole potential can be appoximated as q 1 1 V = 1 ( a/ ) ( a/ )cosθ 1 ( a/ ) ( a/ )cosθ q acosθ pcosθ ˆ = = p whee p = aqˆjis the electic dipole moment. In spheical pola coodinates, the gadient opeato is 1 ˆ 1 = ˆ+ θ+ φ ˆ θ sinθ φ

3 λ 3d+ d λ V( z= 3 d) = ln = ln 3d d The electic potential dieence between the two points is λ 6 V = V( z = 3 d) V( z = 4 d) = ln > 5 Using the act that the electic potential dieence V is equal to the change in potential enegy pe unit chage, we have whee q = e is the chage o the electon. e λ 6 U = q V = ln < 5 (c) I the electon stats out at est at z = 4d then the change in kinetic enegy is 1 K = mv By consevation o enegy, the change in kinetic enegy is e λ 6 K = U = ln > 5 Thus, the magnitude o the velocity at z = 3d is v e λ 6 = ln m Conceptual Questions 1. What is the dieence between electic potential and electic potential enegy?. A uniom electic ield is paallel to the x-axis. In what diection can a chage be displaced in this ield without any extenal wok being done on the chage? 3. Is it sae to stay in an automobile with a metal body duing sevee thundestom? Explain. 5

4 4. Why ae equipotential suaces always pependicula to electic ield lines? 5. The electic ield inside a hollow, uniomly chaged sphee is zeo. Does this imply that the potential is zeo inside the sphee? 3.1 Additional Poblems Cube How much wok is done to assemble eight identical point chages, each o magnitude q, at the cones o a cube o side a? 3.1. Thee Chages 18 6 Thee chages with q = 3. 1 C and q 1 = 6 1 C ae placed on the x-axis, as shown in the igue The distance between q and q1 is a =.6 m. Figue (a) What is the net oce exeted on q by the othe two chages q 1? (b) What is the electic ield at the oigin due to the two chages q 1? (c) What is the electic potential at the oigin due to the two chages q 1? Wok Done on Chages Two chages q 1 = 3. µ C and q = 4. µ C initially ae sepaated by a distance =.cm. An extenal agent moves the chages until they ae = 5.cm apat. (a) How much wok is done by the electic ield in moving the chages om the wok positive o negative? to? Is (b) How much wok is done by the extenal agent in moving the chages om Is the wok positive o negative? to? 6

5 (c) What is the potential enegy o the initial state whee the chages ae apat? =. cm (d) What is the potential enegy o the inal state whee the chages ae = 5.cm apat? (e) What is the change in potential enegy om the initial state to the inal state? Calculating E om V Suppose in some egion o space the electic potential is given by 3 Eaz 3/ V( x, y, z) = V E z+ ( x + y + z ) whee a is a constant with dimensions o length. Find the x, y, and the z-components o the associated electic ield Electic Potential o a Rod A od o length L lies along the x-axis with its let end at the oigin and has a nonuniom chage density λ = α x,whee α is a positive constant. (a) What ae the dimensions o α? (b) Calculate the electic potential at A. Figue 3.1. (c) Calculate the electic potential at point B that lies along the pependicula bisecto o the od a distance b above the x-axis. 7

6 3.1.6 Electic Potential Suppose that the electic potential in some egion o space is given by V( x, y, z) = V exp( k z )cos kx. Find the electic ield eveywhee. Sketch the electic ield lines in the x z plane Calculating Electic Field om the Electic Potential Suppose that the electic potential vaies along the x-axis as shown in Figue below. Figue The potential does not vay in the y- o z -diection. O the intevals shown (ignoe the behavio at the end points o the intevals), detemine the intevals in which has (a) its geatest absolute value. [Ans: 5 V/m in inteval ab.] (b) its least. [Ans: (b) V/m in inteval cd.] (c) Plot E x as a unction o x. (d) What sot o chage distibutions would poduce these kinds o changes in the potential? Whee ae they located? [Ans: sheets o chage extending in the yz diection located at points b, c, d, etc. along the x-axis. Note again that a sheet o chage with chage pe unit aea σ will always poduce a jump in the nomal component o the electic ield o magnitude σ / ε ]. E x 8

7 3.1.8 Electic Potential and Electic Potential Enegy A ight isosceles tiangle o side a has chages q, +q and q aanged on its vetices, as shown in Figue Figue (a) What is the electic potential at point P, midway between the line connecting the +q and q chages, assuming that V = at ininity? [Ans: q/ πε o a.] (b) What is the potential enegy U o this coniguation o thee chages? What is the signiicance o the sign o you answe? [Ans: q /4 πε o a, the negative sign means that wok was done on the agent who assembled these chages in moving them in om ininity.] (c) A outh chage with chage +3q is slowly moved in om ininity to point P. How much wok must be done in this pocess? What is the signiicance o the sign o you answe? [Ans: +3q / πε o a, the positive sign means that wok was done by the agent who moved this chage in om ininity.] Electic Field, Potential and Enegy Thee chages, +5Q, 5Q, and +3Q ae located on the y-axis at y = +4a, y =, and y = 4a, espectively. The point P is on the x-axis at x = 3a. (a) How much enegy did it take to assemble these chages? (b) What ae the x, y, and z components o the electic ield E at P? (c) What is the electic potential V at point P, taking V = at ininity? (d) A outh chage o +Q is bought to P om ininity. What ae the x, y, and z components o the oce F that is exeted on it by the othe thee chages? (e) How much wok was done (by the extenal agent) in moving the outh chage +Q om ininity to P? 9

ELECTROSTATICS::BHSEC MCQ 1. A. B. C. D.

ELECTROSTATICS::BHSEC MCQ 1. A. B. C. D. ELETROSTATIS::BHSE 9-4 MQ. A moving electic chage poduces A. electic field only. B. magnetic field only.. both electic field and magnetic field. D. neithe of these two fields.. both electic field and magnetic