a) Potential in A: -2700 V
b) Potential difference: -26,800 V
c) Work: ![4.3\cdot 10^{-15} J](https://tex.z-dn.net/?f=4.3%5Ccdot%2010%5E%7B-15%7D%20J)
Explanation:
a)
The electric potential at a distance r from a single-point charge is given by:
![V(r)=\frac{kq}{r}](https://tex.z-dn.net/?f=V%28r%29%3D%5Cfrac%7Bkq%7D%7Br%7D)
where
is the Coulomb's constant
q is the charge
r is the distance from the charge
In this problem, we have a system of two charges, so the total potential at a certain point will be given by the algebraic sum of the two potentials.
Charge 1 is
and is located at the origin (x=0, y=0)
Charge 2 is
![q_2=-3.00 \mu C=-3.00\cdot 10^{-6}C](https://tex.z-dn.net/?f=q_2%3D-3.00%20%5Cmu%20C%3D-3.00%5Ccdot%2010%5E%7B-6%7DC)
and is located at (x=0, y = 0.40 m)
Point A is located at (x = 0.40 m, y = 0)
The distance of point A from charge 1 is
![r_{1A}=0.40 m](https://tex.z-dn.net/?f=r_%7B1A%7D%3D0.40%20m)
So the potential due to charge 2 is
![V_1=\frac{(8.99\cdot 10^9)(+2.00\cdot 10^{-6})}{0.40}=+4.50\cdot 10^4 V](https://tex.z-dn.net/?f=V_1%3D%5Cfrac%7B%288.99%5Ccdot%2010%5E9%29%28%2B2.00%5Ccdot%2010%5E%7B-6%7D%29%7D%7B0.40%7D%3D%2B4.50%5Ccdot%2010%5E4%20V)
The distance of point A from charge 2 is
![r_{2A}=\sqrt{0.40^2+0.40^2}=0.566 m](https://tex.z-dn.net/?f=r_%7B2A%7D%3D%5Csqrt%7B0.40%5E2%2B0.40%5E2%7D%3D0.566%20m)
So the potential due to charge 1 is
![V_2=\frac{(8.99\cdot 10^9)(-3.00\cdot 10^{-6})}{0.566}=-4.77\cdot 10^4 V](https://tex.z-dn.net/?f=V_2%3D%5Cfrac%7B%288.99%5Ccdot%2010%5E9%29%28-3.00%5Ccdot%2010%5E%7B-6%7D%29%7D%7B0.566%7D%3D-4.77%5Ccdot%2010%5E4%20V)
Therefore, the net potential at point A is
![V_A=V_1+V_2=+4.50\cdot 10^4 - 4.77\cdot 10^4=-2700 V](https://tex.z-dn.net/?f=V_A%3DV_1%2BV_2%3D%2B4.50%5Ccdot%2010%5E4%20-%204.77%5Ccdot%2010%5E4%3D-2700%20V)
b)
Here we have to calculate the net potential at point B, located at
(x = 0.40 m, y = 0.30 m)
The distance of charge 1 from point B is
![r_{1B}=\sqrt{(0.40)^2+(0.30)^2}=0.50 m](https://tex.z-dn.net/?f=r_%7B1B%7D%3D%5Csqrt%7B%280.40%29%5E2%2B%280.30%29%5E2%7D%3D0.50%20m)
So the potential due to charge 1 at point B is
![V_1=\frac{(8.99\cdot 10^9)(+2.00\cdot 10^{-6})}{0.50}=+3.60\cdot 10^4 V](https://tex.z-dn.net/?f=V_1%3D%5Cfrac%7B%288.99%5Ccdot%2010%5E9%29%28%2B2.00%5Ccdot%2010%5E%7B-6%7D%29%7D%7B0.50%7D%3D%2B3.60%5Ccdot%2010%5E4%20V)
The distance of charge 2 from point B is
![r_{2B}=\sqrt{(0.40)^2+(0.40-0.30)^2}=0.412 m](https://tex.z-dn.net/?f=r_%7B2B%7D%3D%5Csqrt%7B%280.40%29%5E2%2B%280.40-0.30%29%5E2%7D%3D0.412%20m)
So the potential due to charge 2 at point B is
![V_2=\frac{(8.99\cdot 10^9)(-3.00\cdot 10^{-6})}{0.412}=-6.55\cdot 10^4 V](https://tex.z-dn.net/?f=V_2%3D%5Cfrac%7B%288.99%5Ccdot%2010%5E9%29%28-3.00%5Ccdot%2010%5E%7B-6%7D%29%7D%7B0.412%7D%3D-6.55%5Ccdot%2010%5E4%20V)
Therefore, the net potential at point B is
![V_B=V_1+V_2=+3.60\cdot 10^4 -6.55\cdot 10^4 = -29,500 V](https://tex.z-dn.net/?f=V_B%3DV_1%2BV_2%3D%2B3.60%5Ccdot%2010%5E4%20-6.55%5Ccdot%2010%5E4%20%3D%20-29%2C500%20V)
So the potential difference is
![V_B-V_A=-29,500 V-(-2700 V)=-26,800 V](https://tex.z-dn.net/?f=V_B-V_A%3D-29%2C500%20V-%28-2700%20V%29%3D-26%2C800%20V)
c)
The work required to move a charged particle across a potential difference is equal to its change of electric potential energy, and it is given by
![W=q\Delta V](https://tex.z-dn.net/?f=W%3Dq%5CDelta%20V)
where
q is the charge of the particle
is the potential difference
In this problem, we have:
is the charge of the electron
is the potential difference
Therefore, the work required on the electron is
![W=(-1.6\cdot 10^{-19})(-26,800)=4.3\cdot 10^{-15} J](https://tex.z-dn.net/?f=W%3D%28-1.6%5Ccdot%2010%5E%7B-19%7D%29%28-26%2C800%29%3D4.3%5Ccdot%2010%5E%7B-15%7D%20J)