Answer:
Part A:
![E_{midpoint}=0](https://tex.z-dn.net/?f=E_%7Bmidpoint%7D%3D0)
Part B:
![E_{center}=2711.7558 N/C](https://tex.z-dn.net/?f=E_%7Bcenter%7D%3D2711.7558%20N%2FC)
Explanation:
Part A:
Formula of Electric Field Strength:
![E=\frac{1}{4\pi\epsilon}\frac{xQ}{(x^2+R^2)^{3/2}}](https://tex.z-dn.net/?f=E%3D%5Cfrac%7B1%7D%7B4%5Cpi%5Cepsilon%7D%5Cfrac%7BxQ%7D%7B%28x%5E2%2BR%5E2%29%5E%7B3%2F2%7D%7D)
Where:
x is the distance from the ring
R is the radius of the ring
is constant permittivity of free space=8.854*10^-12 farads/meter
Q is the charge
For right Ring E at the midpoint can be calculated as:
x for right plate=25/2=12.5 cm=0.125 m
Radius=R=10/2=5 cm=0.05 m
![E_{right}=\frac{1}{4\pi8.854*10^{-12}}\frac{(0.125)*(20*10^{-19})}{((0.125)^2+(0.05)^2)^{3/2}}\\E_{right}=9208.1758 N/C](https://tex.z-dn.net/?f=E_%7Bright%7D%3D%5Cfrac%7B1%7D%7B4%5Cpi8.854%2A10%5E%7B-12%7D%7D%5Cfrac%7B%280.125%29%2A%2820%2A10%5E%7B-19%7D%29%7D%7B%28%280.125%29%5E2%2B%280.05%29%5E2%29%5E%7B3%2F2%7D%7D%5C%5CE_%7Bright%7D%3D9208.1758%20N%2FC)
For Left Ring E at the midpoint can be calculated as:
Since charge on both plates is +ve and same in magnitude, the electric field will be same for both plates.
![E_{left}=\frac{1}{4\pi8.854*10^{-12}}\frac{(0.125)*(20*10^{-19})}{((0.125)^2+(0.05)^2)^{3/2}}\\E_{left}=9208.1758 N/C](https://tex.z-dn.net/?f=E_%7Bleft%7D%3D%5Cfrac%7B1%7D%7B4%5Cpi8.854%2A10%5E%7B-12%7D%7D%5Cfrac%7B%280.125%29%2A%2820%2A10%5E%7B-19%7D%29%7D%7B%28%280.125%29%5E2%2B%280.05%29%5E2%29%5E%7B3%2F2%7D%7D%5C%5CE_%7Bleft%7D%3D9208.1758%20N%2FC)
Electric Field at midpoint:
Both rings have same magnitude but the direction of fields will be opposite as they have same charge on them.
![E_{midpoint}=E_{left}-E_{right}\\E_{midpoint}=9208.1758-9208.1758\\E_{midpoint}=0](https://tex.z-dn.net/?f=E_%7Bmidpoint%7D%3DE_%7Bleft%7D-E_%7Bright%7D%5C%5CE_%7Bmidpoint%7D%3D9208.1758-9208.1758%5C%5CE_%7Bmidpoint%7D%3D0)
Part B:
At center of left ring:
Due to left ring Electric field at center is zero because x=0.
![E_{left}=\frac{1}{4\pi8.854*10^{-12}}\frac{(0)*(20*10^{-19})}{((0)^2+(0.05)^2)^{3/2}}\\E_{left}=0 N/C](https://tex.z-dn.net/?f=E_%7Bleft%7D%3D%5Cfrac%7B1%7D%7B4%5Cpi8.854%2A10%5E%7B-12%7D%7D%5Cfrac%7B%280%29%2A%2820%2A10%5E%7B-19%7D%29%7D%7B%28%280%29%5E2%2B%280.05%29%5E2%29%5E%7B3%2F2%7D%7D%5C%5CE_%7Bleft%7D%3D0%20N%2FC)
Due to right ring Electric field at center of left ring:
Now: x=25 cm= o.25 m (To the center of left ring)
![E_{right}=\frac{1}{4\pi8.854*10^{-12}}\frac{(0.25)*(20*10^{-19})}{((0.25)^2+(0.05)^2)^{3/2}}\\E_{right}=2711.7558 N/C](https://tex.z-dn.net/?f=E_%7Bright%7D%3D%5Cfrac%7B1%7D%7B4%5Cpi8.854%2A10%5E%7B-12%7D%7D%5Cfrac%7B%280.25%29%2A%2820%2A10%5E%7B-19%7D%29%7D%7B%28%280.25%29%5E2%2B%280.05%29%5E2%29%5E%7B3%2F2%7D%7D%5C%5CE_%7Bright%7D%3D2711.7558%20N%2FC)
Electric Field Strength at center of left ring is same as that of right ring.
![E_{center}=2711.7558 N/C](https://tex.z-dn.net/?f=E_%7Bcenter%7D%3D2711.7558%20N%2FC)