How much positive charge should be given to the earth so as to have the same potential as that of a positively charged sphere of 1 μC and radius 1 cm . (Radius of earth =6400 km )
Question
How much positive charge should be given to the earth so as to have the same potential as that of a positively charged sphere of 1 μC and radius 1 cm . (Radius of earth =6400 km )
Solution
The potential V of a sphere with charge Q and radius r is given by the formula:
V = kQ/r
where k is Coulomb's constant (approximately 9 x 10^9 N m^2/C^2).
Given that the sphere and the Earth have the same potential, we can set up the following equation:
kQ_sphere/r_sphere = kQ_earth/r_earth
We can solve this equation for Q_earth, the charge we need to give to the Earth:
Q_earth = Q_sphere * (r_earth/r_sphere)
Substituting the given values:
Q_earth = 1 μC * (6400 km / 1 cm)
Note that we need to convert the units so they match. There are 1 x 10^5 cm in a km, so:
Q_earth = 1 μC * (6400 x 10^5 cm / 1 cm)
Q_earth = 6400 x 10^5 μC
Therefore, we would need to give the Earth a positive charge of 6400 x 10^5 μC to have the same potential as the sphere.
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