(a) The potential on the surface of a charged sphere of radius R is equal to
![V(R) = k_e \frac{Q}{R}](https://tex.z-dn.net/?f=V%28R%29%20%3D%20k_e%20%20%5Cfrac%7BQ%7D%7BR%7D%20)
where
![k_e = 8.99 \cdot 10^9 N m^2 C^{-2}](https://tex.z-dn.net/?f=k_e%20%3D%208.99%20%5Ccdot%2010%5E9%20N%20m%5E2%20C%5E%7B-2%7D)
is the Coulomb's constant,
![Q](https://tex.z-dn.net/?f=Q%20)
is the charge on the sphere's surface.
For the generator mentioned in the problem, the charge is
![Q= 5 mC=5 \cdot 10^{-3} C](https://tex.z-dn.net/?f=Q%3D%205%20mC%3D5%20%5Ccdot%2010%5E%7B-3%7D%20C)
, while the radius is
![R= \frac{d}{2}= \frac{2.0 m}{2} =1.0 m](https://tex.z-dn.net/?f=R%3D%20%5Cfrac%7Bd%7D%7B2%7D%3D%20%5Cfrac%7B2.0%20m%7D%7B2%7D%20%3D1.0%20m%20)
. Using these values in the formula, we can calculate the potential at the surface:
![V(R)=8.99 \cdot 10^9 N m^2 C^{-2} \frac{5 \cdot 10^{-3} C}{1.0 m}=4.5 \cdot 10^7 V](https://tex.z-dn.net/?f=V%28R%29%3D8.99%20%5Ccdot%2010%5E9%20N%20m%5E2%20C%5E%7B-2%7D%20%20%5Cfrac%7B5%20%5Ccdot%2010%5E%7B-3%7D%20C%7D%7B1.0%20m%7D%3D4.5%20%5Ccdot%2010%5E7%20V%20)
(b) The potential generated by the sphere at a certain distance r from the centre of the sphere is given by
![V(r) = k_e \frac{Q}{r}](https://tex.z-dn.net/?f=V%28r%29%20%3D%20k_e%20%20%5Cfrac%7BQ%7D%7Br%7D%20)
the problem asks at which distance
![V(r) = 1 mV=1\cdot 10^{-3} V](https://tex.z-dn.net/?f=V%28r%29%20%3D%201%20mV%3D1%5Ccdot%2010%5E%7B-3%7D%20V)
. Substituting in the previous formula we can find the value of r:
![r=k_e \frac{Q}{V(r)}= 8.99 \cdot 10^9 N m^2 C^{-2} \frac{5 \cdot 10^{-3}}{1\cdot 10^{-3} V}=4.5 \cdot 10^{10} m](https://tex.z-dn.net/?f=r%3Dk_e%20%20%5Cfrac%7BQ%7D%7BV%28r%29%7D%3D%208.99%20%5Ccdot%2010%5E9%20N%20m%5E2%20C%5E%7B-2%7D%20%5Cfrac%7B5%20%5Ccdot%2010%5E%7B-3%7D%7D%7B1%5Ccdot%2010%5E%7B-3%7D%20V%7D%3D4.5%20%5Ccdot%2010%5E%7B10%7D%20m)
(c) An oxygen atom with 3 missing electrons has a positive charge of +3e, with e being the elementary charge.
The electric potential energy of a charged particle located at some point with voltage V is
![U=q V](https://tex.z-dn.net/?f=U%3Dq%20V)
where q is the charge of the particle, which is in our case
![q=+3e](https://tex.z-dn.net/?f=q%3D%2B3e)
. So we can calculate the energy of the oxygen atom at the distance found in part b, which corresponds to
![r=4.5 \cdot 10^{10}m](https://tex.z-dn.net/?f=r%3D4.5%20%5Ccdot%2010%5E%7B10%7Dm)
and a voltage of
![V=1 mV](https://tex.z-dn.net/?f=V%3D1%20mV)
: