Answer:
Explanation:
The work done on alpha is given by
Charge on alpha particle is 2 the electron charge
![W=\Delta U=U_{b}-U_{a} \quad \Rightarrow(1)](https://tex.z-dn.net/?f=W%3D%5CDelta%20U%3DU_%7Bb%7D-U_%7Ba%7D%20%5Cquad%20%5CRightarrow%281%29)
The potential at center ![\left[U_{a}\right]](https://tex.z-dn.net/?f=%5Cleft%5BU_%7Ba%7D%5Cright%5D)
The electric potential due to multiples point charges is given by
![U_{a}=k q q_{a} \sum_{i=1}^{4}\left(\frac{1}{r_{i}}\right) \quad \Rightarrow(2)](https://tex.z-dn.net/?f=U_%7Ba%7D%3Dk%20q%20q_%7Ba%7D%20%5Csum_%7Bi%3D1%7D%5E%7B4%7D%5Cleft%28%5Cfrac%7B1%7D%7Br_%7Bi%7D%7D%5Cright%29%20%5Cquad%20%5CRightarrow%282%29)
where
=charge on electron
=charge on alpha particle
But and is equal for all charges so ( 2) is equal to
![U_{a}=\frac{4 k(-e)(2 e)}{d} \quad \Rightarrow(3)](https://tex.z-dn.net/?f=U_%7Ba%7D%3D%5Cfrac%7B4%20k%28-e%29%282%20e%29%7D%7Bd%7D%20%5Cquad%20%5CRightarrow%283%29)
The radius is given by
![d=\frac{1}{2} \sqrt{(10)^{2}+(10)^{2}}=7.071 \mathrm{nC}](https://tex.z-dn.net/?f=d%3D%5Cfrac%7B1%7D%7B2%7D%20%5Csqrt%7B%2810%29%5E%7B2%7D%2B%2810%29%5E%7B2%7D%7D%3D7.071%20%5Cmathrm%7BnC%7D)
Substitution in (3) yields
![U_{a}=\frac{-8 \times 9 \times 10^{9} \times\left(1.6 \times 10^{-19}\right)^{2}}{7.071 \times 10^{-9}}=-2.607 \times 10^{-19} \mathrm{J}](https://tex.z-dn.net/?f=U_%7Ba%7D%3D%5Cfrac%7B-8%20%5Ctimes%209%20%5Ctimes%2010%5E%7B9%7D%20%5Ctimes%5Cleft%281.6%20%5Ctimes%2010%5E%7B-19%7D%5Cright%29%5E%7B2%7D%7D%7B7.071%20%5Ctimes%2010%5E%7B-9%7D%7D%3D-2.607%20%5Ctimes%2010%5E%7B-19%7D%20%5Cmathrm%7BJ%7D)
The potential at mid point ![\left[U_{b}\right]](https://tex.z-dn.net/?f=%5Cleft%5BU_%7Bb%7D%5Cright%5D)
due to lower charges the radius is given by
![r_{t}=\sqrt{L^{2}+(L / 2)^{2}}=\sqrt{10^{2}+5^{2}}=11.18 \mathrm{nm}](https://tex.z-dn.net/?f=r_%7Bt%7D%3D%5Csqrt%7BL%5E%7B2%7D%2B%28L%20%2F%202%29%5E%7B2%7D%7D%3D%5Csqrt%7B10%5E%7B2%7D%2B5%5E%7B2%7D%7D%3D11.18%20%5Cmathrm%7Bnm%7D)
And for upper charges the radius is
![r_{u}=5 \mathrm{nm}](https://tex.z-dn.net/?f=r_%7Bu%7D%3D5%20%5Cmathrm%7Bnm%7D)
So the electric potential is
![U_{b} &=k q q_{a}\left[\frac{2}{r_{l}}+\frac{2}{r_{u}}\right]](https://tex.z-dn.net/?f=U_%7Bb%7D%20%26%3Dk%20q%20q_%7Ba%7D%5Cleft%5B%5Cfrac%7B2%7D%7Br_%7Bl%7D%7D%2B%5Cfrac%7B2%7D%7Br_%7Bu%7D%7D%5Cright%5D)
![=k(-e)(2 e)\left[\frac{2}{r_{1}}+\frac{2}{r_{u}}\right]](https://tex.z-dn.net/?f=%3Dk%28-e%29%282%20e%29%5Cleft%5B%5Cfrac%7B2%7D%7Br_%7B1%7D%7D%2B%5Cfrac%7B2%7D%7Br_%7Bu%7D%7D%5Cright%5D)
![=-4 \times\left(1.6 \times 10^{9}\right)\left(1.6 \times 10^{-19}\right)^{2}\left[\frac{1}{11.18 \times 10^{-9}}+\frac{1}{5.0 \times 10^{-9}}\right]](https://tex.z-dn.net/?f=%3D-4%20%5Ctimes%5Cleft%281.6%20%5Ctimes%2010%5E%7B9%7D%5Cright%29%5Cleft%281.6%20%5Ctimes%2010%5E%7B-19%7D%5Cright%29%5E%7B2%7D%5Cleft%5B%5Cfrac%7B1%7D%7B11.18%20%5Ctimes%2010%5E%7B-9%7D%7D%2B%5Cfrac%7B1%7D%7B5.0%20%5Ctimes%2010%5E%7B-9%7D%7D%5Cright%5D)
![=-2.667 \times 10^{-19} \mathrm{J}](https://tex.z-dn.net/?f=%3D-2.667%20%5Ctimes%2010%5E%7B-19%7D%20%5Cmathrm%7BJ%7D)
The work done on the charge is equal to
![W=U_{b}-U_{a}=-2.667 \times 10^{-19}+2.607 \times 10^{-19}=-6.00 \times 10^{-21} \mathrm{J}](https://tex.z-dn.net/?f=W%3DU_%7Bb%7D-U_%7Ba%7D%3D-2.667%20%5Ctimes%2010%5E%7B-19%7D%2B2.607%20%5Ctimes%2010%5E%7B-19%7D%3D-6.00%20%5Ctimes%2010%5E%7B-21%7D%20%5Cmathrm%7BJ%7D)
The work done on is given by
![W=\Delta U=U_{b}-U_{a} \quad \Rightarrow(1)](https://tex.z-dn.net/?f=W%3D%5CDelta%20U%3DU_%7Bb%7D-U_%7Ba%7D%20%5Cquad%20%5CRightarrow%281%29)
The potential at center ![\left[U_{a}\right]](https://tex.z-dn.net/?f=%5Cleft%5BU_%7Ba%7D%5Cright%5D)
The electric potential due to multiples point charges is given by
![U_{a}=k q q_{a} \sum_{i=1}^{4}\left(\frac{1}{r_{i}}\right) \quad \Rightarrow(2)](https://tex.z-dn.net/?f=U_%7Ba%7D%3Dk%20q%20q_%7Ba%7D%20%5Csum_%7Bi%3D1%7D%5E%7B4%7D%5Cleft%28%5Cfrac%7B1%7D%7Br_%7Bi%7D%7D%5Cright%29%20%5Cquad%20%5CRightarrow%282%29)
But and is equal for all charges so ( 2) is equal to
![U_{a}=\frac{4 k(-e)(2 e)}{d} \quad \Rightarrow(3)](https://tex.z-dn.net/?f=U_%7Ba%7D%3D%5Cfrac%7B4%20k%28-e%29%282%20e%29%7D%7Bd%7D%20%5Cquad%20%5CRightarrow%283%29)
The radius is given by
![d=\frac{1}{2} \sqrt{(10)^{2}+(10)^{2}}=7.071 \mathrm{nC}](https://tex.z-dn.net/?f=d%3D%5Cfrac%7B1%7D%7B2%7D%20%5Csqrt%7B%2810%29%5E%7B2%7D%2B%2810%29%5E%7B2%7D%7D%3D7.071%20%5Cmathrm%7BnC%7D)
Substitution in (3) yields
![U_{a}=\frac{-8 \times 9 \times 10^{9} \times\left(1.6 \times 10^{-19}\right)^{2}}{7.071 \times 10^{-9}}=-2.607 \times 10^{-19} \mathrm{J}](https://tex.z-dn.net/?f=U_%7Ba%7D%3D%5Cfrac%7B-8%20%5Ctimes%209%20%5Ctimes%2010%5E%7B9%7D%20%5Ctimes%5Cleft%281.6%20%5Ctimes%2010%5E%7B-19%7D%5Cright%29%5E%7B2%7D%7D%7B7.071%20%5Ctimes%2010%5E%7B-9%7D%7D%3D-2.607%20%5Ctimes%2010%5E%7B-19%7D%20%5Cmathrm%7BJ%7D)
The potential at mid point ![\left[U_{b}\right]](https://tex.z-dn.net/?f=%5Cleft%5BU_%7Bb%7D%5Cright%5D)
due to lower charges the radius is given by
![r_{t}=\sqrt{L^{2}+(L / 2)^{2}}=\sqrt{10^{2}+5^{2}}=11.18 \mathrm{nm}](https://tex.z-dn.net/?f=r_%7Bt%7D%3D%5Csqrt%7BL%5E%7B2%7D%2B%28L%20%2F%202%29%5E%7B2%7D%7D%3D%5Csqrt%7B10%5E%7B2%7D%2B5%5E%7B2%7D%7D%3D11.18%20%5Cmathrm%7Bnm%7D)
And for upper charges the radius is
![r_{u}=5 \mathrm{nm}](https://tex.z-dn.net/?f=r_%7Bu%7D%3D5%20%5Cmathrm%7Bnm%7D)
So the electric potential is
![U_{b} &=k q q_{a}\left[\frac{2}{r_{l}}+\frac{2}{r_{u}}\right]](https://tex.z-dn.net/?f=U_%7Bb%7D%20%26%3Dk%20q%20q_%7Ba%7D%5Cleft%5B%5Cfrac%7B2%7D%7Br_%7Bl%7D%7D%2B%5Cfrac%7B2%7D%7Br_%7Bu%7D%7D%5Cright%5D)
![=k(-e)(2 e)\left[\frac{2}{r_{1}}+\frac{2}{r_{u}}\right]](https://tex.z-dn.net/?f=%3Dk%28-e%29%282%20e%29%5Cleft%5B%5Cfrac%7B2%7D%7Br_%7B1%7D%7D%2B%5Cfrac%7B2%7D%7Br_%7Bu%7D%7D%5Cright%5D)
![=-4 \times\left(1.6 \times 10^{9}\right)\left(1.6 \times 10^{-19}\right)^{2}\left[\frac{1}{11.18 \times 10^{-9}}+\frac{1}{5.0 \times 10^{-9}}\right]](https://tex.z-dn.net/?f=%3D-4%20%5Ctimes%5Cleft%281.6%20%5Ctimes%2010%5E%7B9%7D%5Cright%29%5Cleft%281.6%20%5Ctimes%2010%5E%7B-19%7D%5Cright%29%5E%7B2%7D%5Cleft%5B%5Cfrac%7B1%7D%7B11.18%20%5Ctimes%2010%5E%7B-9%7D%7D%2B%5Cfrac%7B1%7D%7B5.0%20%5Ctimes%2010%5E%7B-9%7D%7D%5Cright%5D)
![=-2.667 \times 10^{-19} \mathrm{J}](https://tex.z-dn.net/?f=%3D-2.667%20%5Ctimes%2010%5E%7B-19%7D%20%5Cmathrm%7BJ%7D)
The work done on the charge is equal to
![W=U_{b}-U_{a}=-2.667 \times 10^{-19}+2.607 \times 10^{-19}=-6.00 \times 10^{-21} \mathrm{J}](https://tex.z-dn.net/?f=W%3DU_%7Bb%7D-U_%7Ba%7D%3D-2.667%20%5Ctimes%2010%5E%7B-19%7D%2B2.607%20%5Ctimes%2010%5E%7B-19%7D%3D-6.00%20%5Ctimes%2010%5E%7B-21%7D%20%5Cmathrm%7BJ%7D)
![W=-6.00 \times 10^{-21} \mathrm{J}](https://tex.z-dn.net/?f=W%3D-6.00%20%5Ctimes%2010%5E%7B-21%7D%20%5Cmathrm%7BJ%7D)