Categories
Physics Olympiad

Potential of Metal Sphere

Let’s discuss the problem where we have to find the potential of metal sphere.

The Problem:

A metal sphere having a radius (r_1) charged to a potential (\phi_1) is enveloped by a thin-walled conducting spherical shell of radius (r_2). Determine the potential (\phi_2) acquired by the sphere after it has been connected for a short time to the shell by a conductor.

Solution:

The charge (q_1) of the sphere can be determined from the relation $$ q_1=4\pi\epsilon_0r_1$$
After the connection of the sphere to the envelope, the entire charge (q_1) will flow from the sphere to the envelope and will be distributed uniformly over its surface.
Its potential (\phi_2) (coinciding with the new value of the potential of the sphere) will be
$$ \phi_2=\frac{q_1}{4\pi\epsilon_0r_2}=\phi_1\frac{r_1}{r_2}$$

Categories
Physics Olympiad

Total Charge of a Sphere

Try this problem, useful for the Physics Olympiad Problem based on total charge of a sphere.

The Problem:

Suppose a charge (Q) is distributed within a sphere of radius (R) in such a way that the charge density (\rho(r)) at a distance r from the centre of the sphere is
$$ \rho(r)=K(R-r) \hspace{2mm }for\hspace{2mm} 0<r<R$$
$$ 0 \hspace{2mm} for \hspace{2mm} r>R$$

Determine the total charge (Q).
Solution:

Let us consider a thin spherical shell of radius (r) and thickness (dr). Charge within it is (\rho(r).4\pi r^2dr). Therefore, the total charge $$ Q=\int_{0}^{R}\rho(x).4\pi r^2dr$$$$=4\pi K\int_{0}^{R}(R-r)^2dr$$$$=\pi KR^4/3$$

Categories
Physics Olympiad

Total Charge of a Circular Wire

Try this problem, useful for Physics Olympiad based on Total Charge of a Circular Wire.

The Problem:

A circular wire of radius (a) has linear charge density $$\lambda=\lambda_0cos^2\theta$$ where (\theta) is the angle with respect to a fixed radius. Calculate the total charge.

Solution:

Charge on an element (dl=ad\theta) is $$ \lambda_0cos^2\theta.ad\theta$$
Total charge $$ Q=\int_{0}^{2\pi}a\lambda_0cos^2\theta d\theta$$$$=a\lambda_0\pi$$