Question

Three concentric metallic spherical shells of radii r, 2r, 3r, are given charges q1, q2, q3, respectively. it is found that the surface charge densities on the outer surfaces of the shells are equal. then, the ratio of the charges given to the shells, q1 : q2 : q3, is

Answer 1

Explanation:

radius of the shell 1 = r

radius of shell 2 = 2r

radius of shell 3 = 3r

we have to find the ratio  of the charges  of the shell

since surface charge densities on the outer surfaces of the shells are equal

therefore

$Q_1 = \sigma(4\pir^2) = 4\pi\sigma r^2\\\\Q_2 = \sigma(4\pi (2r)^2)= 16\pi\sigma r^2- Q_1 = 12\pi\sigma r^2\\\\Q_3 = \sigma(4\pi (3r)^2)= 36\pi\sigma r^2- Q_2= 20\pi\sigma r^2$

Thus ,

ratio of

$Q_1 :Q_2:Q_3 \\\\=1:3:5$

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Answer 2

Given that,

The radius of three concentric metallic spherical shells are

$R_{1}=r$

$R_{2}=2r$

$R_{3}=3r$

The charges of spherical shells are q₁, q₂ and q₃.

We know that,

The charge density is

$\sigma=\dfrac{q}{4\pi R^2}$

The charge on the first sphere is q₁.

The charge on the second sphere is $q_{1}+q_{2}$

The charge on the third sphere is $q_{1}+q_{2}+q_{3}$

The charge density of first sphere is

$\sigma_{1}=\dfrac{q_{1}}{4\pi r^2}$...(I)

The charge density of second sphere is

$\sigma_{2}=\dfrac{q_{1}+q_{2}}{4\pi 4r^2}$....(II)

The charge density of third sphere is

$\sigma_{3}=\dfrac{q_{1}+q_{2}+q_{3}}{4\pi 9r^2}$.....(III)

The surface charge densities on the outer surfaces of the shells are equal.

We need to calculate the ratio of the charges

Using formula of charge density

$\sigma_{1}=\sigma_{2}=\sigma_{3} = x$

Put the value of charge density

$\dfrac{q_{1}}{4\pi r^2}=\dfrac{q_{1}+q_{2}}{4\pi 4r^2}=\dfrac{q_{1}+q_{2}+q_{3}}{4\pi 9r^2}=x$

From equation (I)

$q_{1}=4\pi r^2x$....(IV)

From equation (II)

$q_{1}+q_{2}=4\pi 4r^2x$

$q_{2}=4\pi 4r^2x-q_{1}$

Put the value of q₁

$q_{2}=4\pi 4r^2-4\pi r^2x$

$q_{2}=4\pi r^2x(4-1)$

$q_{2}=3q_{1}$

Now, from equation (III)

$q_{1}+q_{2}+q_{3}=4\pi 9r^2x$

$q_{3}=4\pi 9r^2x-q_{2}-q_{1}$

Put the value of q₂

$q_{3}=4\pi 9r^2x-3q_{1}-q_{1}$

$q_{3}=9q_{1}-3q_{1}-q_{1}$

$q_{3}=5q_{1}$

The ratio of the charges are

$q_{1}:q_{2}:q_{3}=q_{1}:3q_{1}:5q_{1}$

$q_{1}:q_{2}:q_{3}=1:3:5$

Hence, The ratio of the charges are 1 : 3 : 5