Answer:
Φ = 361872 N.m^2 / C
Explanation:
Given:-
- The area of the two plates, ![A_p = 180 cm^2](https://tex.z-dn.net/?f=A_p%20%3D%20180%20cm%5E2)
- The charge on each plate, ![q = 17 * 10^-^6 C](https://tex.z-dn.net/?f=q%20%3D%2017%20%2A%2010%5E-%5E6%20C)
- Permittivity of free space, ![e_o = 8.85 * 10^-^1^2 \frac{C^2}{N.m^2}](https://tex.z-dn.net/?f=e_o%20%3D%208.85%20%2A%2010%5E-%5E1%5E2%20%5Cfrac%7BC%5E2%7D%7BN.m%5E2%7D)
- The radius for the flux region, ![r = 3.3 cm](https://tex.z-dn.net/?f=r%20%3D%203.3%20cm)
- The angle between normal to region and perpendicular to plates, θ = 4°
Find:-
Find the flux (in N · m2/C) through a circle of radius 3.3 cm between the plates.
Solution:-
- First we will determine the area of the region ( Ar ) by using the formula for the area of a circle as follows. The region has a radius of r = 3.3 cm:
![A_r = \pi *r^2\\\\A_r = \pi *(0.033)^2\\\\A_r = 0.00342 m^2](https://tex.z-dn.net/?f=A_r%20%3D%20%5Cpi%20%2Ar%5E2%5C%5C%5C%5CA_r%20%3D%20%5Cpi%20%2A%280.033%29%5E2%5C%5C%5C%5CA_r%20%3D%200.00342%20m%5E2)
- The charge density ( σ ) would be considered to be uniform for both plates. It is expressed as the ratio of the charge ( q ) on each plate and its area ( A_p ):
σ = ![\frac{q}{A_p} = \frac{17*10^-^6}{0.018} \\](https://tex.z-dn.net/?f=%5Cfrac%7Bq%7D%7BA_p%7D%20%3D%20%5Cfrac%7B17%2A10%5E-%5E6%7D%7B0.018%7D%20%5C%5C)
σ = 0.00094 C / m^2
- We will assume the electric field due to the positive charged plate ( E+ ) / negative charged plate ( E- ) to be equivalent to the electric field ( E ) of an infinitely large charged plate with uniform charge density.
![E+ = E- = \frac{sigma}{2*e_o} \\\\](https://tex.z-dn.net/?f=E%2B%20%3D%20E-%20%3D%20%5Cfrac%7Bsigma%7D%7B2%2Ae_o%7D%20%5C%5C%5C%5C)
- The electric field experienced by a region between two infinitely long charged plates with uniform charge density is the resultant effect of both plates. So from the principle of super-position we have the following net uniform electric field ( E_net ) between the two plates:
![E_n_e_t = (E+) + ( E-)\\\\E_n_e_t = \frac{0.00094}{8.85*10^-^1^2} \\\\E_n_e_t = 106214689.26553 \frac{N}{C} \\](https://tex.z-dn.net/?f=E_n_e_t%20%3D%20%28E%2B%29%20%20%2B%20%28%20E-%29%5C%5C%5C%5CE_n_e_t%20%3D%20%5Cfrac%7B0.00094%7D%7B8.85%2A10%5E-%5E1%5E2%7D%20%5C%5C%5C%5CE_n_e_t%20%3D%20106214689.26553%20%5Cfrac%7BN%7D%7BC%7D%20%20%5C%5C)
- From the Gauss-Law the flux ( Φ ) through a region under uniform electric field ( E_net ) at an angle of ( θ ) is:
Φ = E_net * Ar * cos ( θ )
Φ = (106214689.26553) * (0.00342) * cos ( 5 )
Φ = 361872 N.m^2 / C