Two identical point charges are separated by a distance of 0.286 m. Exactly half

Question

Two identical point charges are separated by a distance of 0.286 m. Exactly halfway between the two charge! the electric potential is -4.72 times 10s V (taking the potential to be zero at infinity). What is the value of the two charges? (Similar to problem 23.14 & 23.23) A solid conducting sphere has a radius of 12.6cm. Taking the potential at infinity to be zero. the electric potential 3.80 cm from the sphere's center is 471 V (Similar to problem 23.42) Does the sphere haw too many electrons or too few? How many electrons? A hollow conducting sphere has a radius of 5 00 cm and a net electrical charge of 5.G0 mC. The hollow sphere is fixed In position A small, solid sphere with a radius of 0.230 mm and a mass

Explanation / Answer

1) The given to you is that the total potential is -4.72*10^5 V. If you solve for each potential you can get the charges.
Use the equation V_tot= V_1+V_2
Use the integration for V for the points (,r], so V_1= -Edr, since the points are separated one dimensionally the E vector is the same as the magnitude of E, so E= kq/r^2

V_1= -kq/r^2dr

V_1= -kq1/r^2 dr= kq/r, if you use the points (,r) you get kq/r

Since the point is exactly in the middle, and the charges are identical V_2=V_1

So

V_tot=V_1+V_2= kq/r+kq/r

V_tot= 2kq/r, q=(V_tot*r)/2k, where r is half of the distance between the charges, and k is the constant

q= -3.75*10^-6 C, or -3.75C

2) Part two is using the same change in potential formula using the value E= kq/r^2, Also to point out since the surface your using is a conductor there is no E field inside the sphere so there is no potential inside the conductor

a) Im guessing they are asking for the sign of the charge, so if it's positive there are too few electorns and if negative there are too many. Also the only voltage applies from the points to r, the radius of the conductor.

V_rR= -kq/r^2dr (,r]-0

V_r= kq/r

q= (r*V_r)/k, r= .126m

q= (471*.126)/8.99*10^9= 6.60*10^-9= 6.60nC, so too few electrons

b) just divide the total charge of the sphere by the absolute value of the charge to get the number of electrons needed.

x(q_e)= q

x=q/q_e

x= 6.60*10^-9/1.60× 10-19

x= 4.13*10^10 electrons

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