<h2>20. How much charge is on sphere B after A and B touch and are separated?</h2><h3>Answer:</h3>
![\boxed{q_{B}=+2q}](https://tex.z-dn.net/?f=%5Cboxed%7Bq_%7BB%7D%3D%2B2q%7D)
<h3>Explanation:</h3>
We'll solve this problem by using the concept of electric potential or simply called potential
, which is <em>the energy per unit charge, </em>so the potential
at any point in an electric field with a test charge
at that point is:
![V=\frac{U}{q_{0}}](https://tex.z-dn.net/?f=V%3D%5Cfrac%7BU%7D%7Bq_%7B0%7D%7D)
The potential
due to a single point charge q is:
![V=k\frac{q}{r}](https://tex.z-dn.net/?f=V%3Dk%5Cfrac%7Bq%7D%7Br%7D)
Where
is an electric constant,
is value of point charge and
is the distance from point charge to where potential is measured. Since, the three spheres A, B and C are identical, they have the same radius
. Before the sphere A and B touches we have:
![V_{A}=k\frac{q_{A}}{r_{A}} \\ \\ V_{B}=k\frac{q_{B}}{r_{A}} \\ \\ But: \\ \\ \ r_{A}=r_{B}=r](https://tex.z-dn.net/?f=V_%7BA%7D%3Dk%5Cfrac%7Bq_%7BA%7D%7D%7Br_%7BA%7D%7D%20%5C%5C%20%5C%5C%20V_%7BB%7D%3Dk%5Cfrac%7Bq_%7BB%7D%7D%7Br_%7BA%7D%7D%20%5C%5C%20%5C%5C%20But%3A%20%5C%5C%20%5C%5C%20%5C%20r_%7BA%7D%3Dr_%7BB%7D%3Dr)
When they touches each other the potential is the same, so:
![V_{A}= V_{B} \\ \\ k\frac{q_{A}}{r}=k\frac{q_{B}}{r} \\ \\ \boxed{q_{A}=q_{B}}](https://tex.z-dn.net/?f=V_%7BA%7D%3D%20V_%7BB%7D%20%5C%5C%20%5C%5C%20k%5Cfrac%7Bq_%7BA%7D%7D%7Br%7D%3Dk%5Cfrac%7Bq_%7BB%7D%7D%7Br%7D%20%5C%5C%20%5C%5C%20%5Cboxed%7Bq_%7BA%7D%3Dq_%7BB%7D%7D)
From the principle of conservation of charge <em>the algebraic sum of all the electric charges in any closed system is constant. </em>So:
![q_{A}+q_{B}=q \\ \\ q_{A}=+6q \ and \ q_{B}=-2q \\ \\ So: \\ \\ \boxed{q_{A}+q_{B}=+4q}](https://tex.z-dn.net/?f=q_%7BA%7D%2Bq_%7BB%7D%3Dq%20%5C%5C%20%5C%5C%20q_%7BA%7D%3D%2B6q%20%5C%20and%20%5C%20q_%7BB%7D%3D-2q%20%5C%5C%20%5C%5C%20So%3A%20%5C%5C%20%5C%5C%20%5Cboxed%7Bq_%7BA%7D%2Bq_%7BB%7D%3D%2B4q%7D)
Therefore:
![(1) \ q_{A}=q_{B} \\ \\ (2) \ q_{A}+q_{B}=+4q \\ \\ (1) \ into \ (2): \\ \\ q_{A}+q_{A}=+4q \therefore 2q_{A}=+4q \therefore \boxed{q_{A}=q_{B}=+2q}](https://tex.z-dn.net/?f=%281%29%20%5C%20q_%7BA%7D%3Dq_%7BB%7D%20%5C%5C%20%5C%5C%20%282%29%20%5C%20q_%7BA%7D%2Bq_%7BB%7D%3D%2B4q%20%5C%5C%20%5C%5C%20%281%29%20%5C%20into%20%5C%20%282%29%3A%20%5C%5C%20%5C%5C%20q_%7BA%7D%2Bq_%7BA%7D%3D%2B4q%20%5Ctherefore%202q_%7BA%7D%3D%2B4q%20%5Ctherefore%20%5Cboxed%7Bq_%7BA%7D%3Dq_%7BB%7D%3D%2B2q%7D)
So after A and B touch and are separated the charge on sphere B is:
![\boxed{q_{B}=+2q}](https://tex.z-dn.net/?f=%5Cboxed%7Bq_%7BB%7D%3D%2B2q%7D)
<h2>21. How much charge ends up on sphere C?</h2><h3>Answer:</h3>
![\boxed{q_{C}=+1.5q}](https://tex.z-dn.net/?f=%5Cboxed%7Bq_%7BC%7D%3D%2B1.5q%7D)
<h3>Explanation:</h3>
First: A and B touches and are separated, so the charges are:
![q_{A}=q_{B}=+2q](https://tex.z-dn.net/?f=q_%7BA%7D%3Dq_%7BB%7D%3D%2B2q)
Second: C is then touched to sphere A and separated from it.
Third: C is to sphere B and separated from it
So we need to calculate the charge that ends up on sphere C at the third step, so we also need to calculate step second. Therefore, from the second step:
Here
and C carries no net charge or
. Also, ![r_{A}=r_{C}=r](https://tex.z-dn.net/?f=r_%7BA%7D%3Dr_%7BC%7D%3Dr)
![V_{A}=k\frac{q_{A}}{r} \\ \\ V_{C}=k\frac{q_{C}}{r}](https://tex.z-dn.net/?f=V_%7BA%7D%3Dk%5Cfrac%7Bq_%7BA%7D%7D%7Br%7D%20%5C%5C%20%5C%5C%20V_%7BC%7D%3Dk%5Cfrac%7Bq_%7BC%7D%7D%7Br%7D)
Applying the same concept as the previous problem when sphere touches we have:
![k\frac{q_{A}}{r} =k\frac{q_{C}}{r} \\ \\ q_{A}=q_{C}](https://tex.z-dn.net/?f=k%5Cfrac%7Bq_%7BA%7D%7D%7Br%7D%20%3Dk%5Cfrac%7Bq_%7BC%7D%7D%7Br%7D%20%5C%5C%20%5C%5C%20q_%7BA%7D%3Dq_%7BC%7D)
For the principle of conservation of charge:
![q_{A}+q_{C}=+2q \\ \\ q_{A}=q_{C}=+q](https://tex.z-dn.net/?f=q_%7BA%7D%2Bq_%7BC%7D%3D%2B2q%20%5C%5C%20%5C%5C%20q_%7BA%7D%3Dq_%7BC%7D%3D%2Bq)
Finally, from the third step:
Here
. Also, ![r_{B}=r_{C}=r](https://tex.z-dn.net/?f=r_%7BB%7D%3Dr_%7BC%7D%3Dr)
![V_{B}=k\frac{q_{B}}{r} \\ \\ V_{C}=k\frac{q_{C}}{r}](https://tex.z-dn.net/?f=V_%7BB%7D%3Dk%5Cfrac%7Bq_%7BB%7D%7D%7Br%7D%20%5C%5C%20%5C%5C%20V_%7BC%7D%3Dk%5Cfrac%7Bq_%7BC%7D%7D%7Br%7D)
When sphere touches we have:
![k\frac{q_{B}}{r} =k\frac{q_{C}}{r} \\ \\ q_{B}=q_{C}](https://tex.z-dn.net/?f=k%5Cfrac%7Bq_%7BB%7D%7D%7Br%7D%20%3Dk%5Cfrac%7Bq_%7BC%7D%7D%7Br%7D%20%5C%5C%20%5C%5C%20q_%7BB%7D%3Dq_%7BC%7D)
For the principle of conservation of charge:
![q_{B}+q_{C}=+3q \\ \\ q_{A}=q_{C}=+1.5q](https://tex.z-dn.net/?f=q_%7BB%7D%2Bq_%7BC%7D%3D%2B3q%20%5C%5C%20%5C%5C%20q_%7BA%7D%3Dq_%7BC%7D%3D%2B1.5q)
So the charge that ends up on sphere C is:
![q_{C}=+1.5q](https://tex.z-dn.net/?f=q_%7BC%7D%3D%2B1.5q)
<h2>
22. What is the total charge on the three spheres before they are allowed to touch each other.</h2><h3>Answer:</h3>
![+4q](https://tex.z-dn.net/?f=%2B4q)
<h3>Explanation:</h3>
Before they are allowed to touch each other we have that:
![q_{A}=+6q \\ \\ q_{B}=-2q \\ \\ q_{C}=0](https://tex.z-dn.net/?f=q_%7BA%7D%3D%2B6q%20%5C%5C%20%5C%5C%20q_%7BB%7D%3D-2q%20%5C%5C%20%5C%5C%20q_%7BC%7D%3D0)
Therefore, for the principle of conservation of charge <em>the algebraic sum of all the electric charges in any closed system is constant, </em>then this can be expressed as:
![q_{A}+q_{B}+q_{C}=+6q -2q +0 \\ \\ \therefore q_{A}+q_{B}+q_{C}=+4q](https://tex.z-dn.net/?f=q_%7BA%7D%2Bq_%7BB%7D%2Bq_%7BC%7D%3D%2B6q%20-2q%20%2B0%20%5C%5C%20%5C%5C%20%5Ctherefore%20q_%7BA%7D%2Bq_%7BB%7D%2Bq_%7BC%7D%3D%2B4q)
Lastly, the total charge on the three spheres before they are allowed to touch each other is:
![+4q](https://tex.z-dn.net/?f=%2B4q)