Question

For what value of the ratio r/a of plate radius to separation between the plates does the electric field at the point x=a/2 on the x axis differ by 1 percent from the result η/ϵ0 for infinite sheets?

Answer 1

The question seeks the ratio r/a that results in the electric field strength at x=a/2, between two finite parallel conducting plates, differing by 1% from that of an ideal infinite plate. The exact electric field needs to be calculated, involving electromagnetism principles and numerical approximation if necessary, and compared to the infinite plate case to find the desired ratio.

Step-by-step explanation:

The question pertains to finding the value of the ratio r/a where the electric field strength at a point x=a/2 on the axis of two parallel conducting plates differs by 1% from the value η/ε0 for a pair of infinite sheets with surface charge density η. This involves applying concepts from electromagnetism, particularly the electric field due to a finite-sized charged plate, and comparing it to the ideal case of an infinite plate.

To solve this problem, we would typically start by using the formula E = σ/2ε0 for the electric field due to an infinite sheet of charge, where σ is the surface charge density and ε0 is the permittivity of free space. Since the plate is of finite size, however, this formula will only approximate the field very close to the plate. To find the actual field at point x=a/2 and compare it with the ideal case, we must use the solution for finite-sized plates, potentially involving numerical methods or approximation techniques such as the method of images.

After finding the exact field strength at x=a/2, we set this value to be 99% of η/ε0 to reflect the 1% deviation specified. Then, we solve for the ratio r/a that satisfies this condition.

Answer 2

E\delta =\frac{R}{\epsilon0}(1-\frac{A}{\sqrt{4R^{2}}-ac}

=\frac{R}{\epsilon0}(1-\frac{1}{\sqrt{4r^{2}/^{_a{2}}+1}})

=\frac{R}{\epsilon0}(1-\frac{1}{\sqrt{4x^2+1}})

x=\frac{r}{a}

infinite case,

Ei=\frac{r}{\epsilon0}

\therefore e\delta =ei(1-\frac{1}{\sqrt{4x^{2}+1}})

we have to find x when,

ei-e\delta =1% ,y=ei=1/100 ei

or,ei-ei+\frac{ei}{\sqrt{4x^2+1}} = 1/100ei

\frac{1}{\sqrt{4x^2+1}}=\frac{1}{100}

4x^2+1 =10^4

x=\frac{\sqrt{\frac{10^4-1}{4}}}=49.99\approx 50

\therefore \frac{r}{a}\approx 50