Name: Dielectrics in a Capacitor
Uploaded: 2019-07-19
Duration: 6 min 3 s
Description: Let us use our knowledge about dielectrics and capacitors to find out how the use of dielectrics can increase the capacitance of a system. Learn through this...

Question

A current

I

flows along a thin wire shaped as a regular polygon with

n

sides, which can be inscribed into a circle of radius

R

. Find the magnetic induction at the centre of the polygon. Analyze the obtained expression at

n \to \infty

.

Accepted Answer

In the triangle $A O B$ ,
the angle $∠ A O B = \frac{2 π}{n}$ .
and $O C$ is the perpendicular distance of any segment from the centre, which equals to $R \cos \frac{π}{n}$ .
Now, the magnetic induction at $O$ , due to the right current-carrying element $A B$ is

$B_{1} = \frac{μ_{0}}{4 π} \frac{i}{R c o s \frac{π}{n}} 2 s i n \frac{π}{n}$ .

From Biot-Savart's law, the magnetic field at $O$ due to any section such as $A B$ is perpendicular to the plane of the figure and has the magnitude

$B = \int \frac{μ_{0}}{4 π} i \frac{d x}{r^{2}} \cos θ = \int_{- \frac{π}{n}}^{\frac{π}{n}} \frac{μ_{0} i}{4 π} \frac{R \cos \frac{π}{n} \sec^{2} θ d θ}{R^{2} \cos^{2} \frac{π}{n} \sec^{2} θ} \cos θ = \frac{μ_{0} i}{4 π} \frac{1}{R \cos \frac{π}{n}} 2 \sin \frac{π}{n}$

There are $n$ number of sides and magnetic induction vectors due to each side at $O$ , which are equal in magnitude and direction. So,
$B_{0} = \frac{μ_{0}}{4 π} \frac{n i}{R \cos \frac{π}{n}} 2 \sin \frac{π}{n} \cdot n$
$= \frac{μ_{0}}{2 π} \frac{n i}{R} \tan \frac{π}{n}$ .

Now, for $n \to \infty$ ,
$B_{0} = \frac{μ_{0}}{2} \frac{i}{R} \underset{n \to \infty}{L t} (\frac{\tan \frac{π}{n}}{\frac{π}{n}}) = \frac{μ_{0}}{2} \frac{i}{R}$ the same as that due to a circular loop.

A current $I$ flows along a thin wire shaped as a regular polygon with $n$ sides, which can be inscribed into a circle of radius $R$ . Find the magnetic induction at the centre of the polygon. Analyze the obtained expression at $n \to \infty$ .

Important Questions on ELECTRODYNAMICS

Learn and Prepare for any exam you want !

A current I flows along a thin wire shaped as a regular polygon with n sides, which can be inscribed into a circle of radius R. Find the magnetic induction at the centre of the polygon. Analyze the obtained expression at n→∞.

Important Questions on ELECTRODYNAMICS

Learn and Prepare for any exam you want !

A current $I$ flows along a thin wire shaped as a regular polygon with $n$ sides, which can be inscribed into a circle of radius $R$ . Find the magnetic induction at the centre of the polygon. Analyze the obtained expression at $n \to \infty$ .