Answer:
(a). If z = 0, The electric field due to the rod is zero.
(b). If z = ∞, The electric field due to the rod is
.
(c). The positive distance is ![\dfrac{R}{\sqrt{2}}](https://tex.z-dn.net/?f=%5Cdfrac%7BR%7D%7B%5Csqrt%7B2%7D%7D)
(d). The maximum magnitude of electric field is ![1.54\times10^{4}\ N/C](https://tex.z-dn.net/?f=1.54%5Ctimes10%5E%7B4%7D%5C%20N%2FC)
Explanation:
Given that,
Radius = 2.00 cm
Charge = 4.00 mC
(a). If the radius and charge are R and Q.
We need to calculate the electric field due to the rod
Using formula of electric field
![E=\dfrac{1}{4\pi\epsilon_{0}}\dfrac{Qz}{(z^2+R^2)^{\frac{2}{3}}}](https://tex.z-dn.net/?f=E%3D%5Cdfrac%7B1%7D%7B4%5Cpi%5Cepsilon_%7B0%7D%7D%5Cdfrac%7BQz%7D%7B%28z%5E2%2BR%5E2%29%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D%7D)
Where, Q = charge
z = distance
If z = 0,
Then, The electric field is
![E=0](https://tex.z-dn.net/?f=E%3D0)
(b). If z = ∞, z>>R
So, R = 0
Then, the electric field is
![E=\dfrac{1}{4\pi\epsilon_{0}}\dfrac{Q}{z^2}](https://tex.z-dn.net/?f=E%3D%5Cdfrac%7B1%7D%7B4%5Cpi%5Cepsilon_%7B0%7D%7D%5Cdfrac%7BQ%7D%7Bz%5E2%7D)
![E\propto\dfrac{1}{z^2}](https://tex.z-dn.net/?f=E%5Cpropto%5Cdfrac%7B1%7D%7Bz%5E2%7D)
(c). In terms of R,
We need to calculate the positive distance
If ![E\rightarrow E_{max}](https://tex.z-dn.net/?f=E%5Crightarrow%20E_%7Bmax%7D)
Then, ![\dfrac{dE}{dz}=0](https://tex.z-dn.net/?f=%5Cdfrac%7BdE%7D%7Bdz%7D%3D0)
![\dfrac{Q}{4\pi\epsilon_{0}}(\dfrac{(z^2+R^2)^\frac{3}{2}-\dfrac{3z}{2}(z^2+R^2)^\dfrac{1}{2}}{(z^2+R^2)^2})=0](https://tex.z-dn.net/?f=%5Cdfrac%7BQ%7D%7B4%5Cpi%5Cepsilon_%7B0%7D%7D%28%5Cdfrac%7B%28z%5E2%2BR%5E2%29%5E%5Cfrac%7B3%7D%7B2%7D-%5Cdfrac%7B3z%7D%7B2%7D%28z%5E2%2BR%5E2%29%5E%5Cdfrac%7B1%7D%7B2%7D%7D%7B%28z%5E2%2BR%5E2%29%5E2%7D%29%3D0)
Taking only positive distance
![z=\dfrac{R}{\sqrt{2}}](https://tex.z-dn.net/?f=z%3D%5Cdfrac%7BR%7D%7B%5Csqrt%7B2%7D%7D)
(d). If R = 2.00 and Q = 4.00 mC
We need to calculate the maximum magnitude of electric field
Using formula of electric field
![E_{max}=\dfrac{1}{4\pi\epsilon_{0}}\dfrac{Qz}{(z^2+R^2)^{\frac{2}{3}}}](https://tex.z-dn.net/?f=E_%7Bmax%7D%3D%5Cdfrac%7B1%7D%7B4%5Cpi%5Cepsilon_%7B0%7D%7D%5Cdfrac%7BQz%7D%7B%28z%5E2%2BR%5E2%29%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D%7D)
![E_{max}=9\times10^{9}\times\dfrac{4.0\times10^{-6}\times\dfrac{2.00}{\sqrt{2}}}{((\dfrac{2.00}{\sqrt{2}})^2+(2.00)^2)^{\frac{2}{3}}}](https://tex.z-dn.net/?f=E_%7Bmax%7D%3D9%5Ctimes10%5E%7B9%7D%5Ctimes%5Cdfrac%7B4.0%5Ctimes10%5E%7B-6%7D%5Ctimes%5Cdfrac%7B2.00%7D%7B%5Csqrt%7B2%7D%7D%7D%7B%28%28%5Cdfrac%7B2.00%7D%7B%5Csqrt%7B2%7D%7D%29%5E2%2B%282.00%29%5E2%29%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D%7D)
![E_{max}=15418.7\ N/C](https://tex.z-dn.net/?f=E_%7Bmax%7D%3D15418.7%5C%20N%2FC)
![E_{max}=1.54\times10^{4}\ N/C](https://tex.z-dn.net/?f=E_%7Bmax%7D%3D1.54%5Ctimes10%5E%7B4%7D%5C%20N%2FC)
Hence, (a). If z = 0, The electric field due to the rod is zero.
(b). If z = ∞, The electric field due to the rod is
.
(c). The positive distance is ![\dfrac{R}{\sqrt{2}}](https://tex.z-dn.net/?f=%5Cdfrac%7BR%7D%7B%5Csqrt%7B2%7D%7D)
(d). The maximum magnitude of electric field is ![1.54\times10^{4}\ N/C](https://tex.z-dn.net/?f=1.54%5Ctimes10%5E%7B4%7D%5C%20N%2FC)