can a metal sphere of radius 1cm hold a charge of 1 coulomb?justify your answer.
To determine if a metal sphere of radius 1cm can hold a charge of 1 coulomb, we need to consider a few factors: the capacitance of the sphere, the electric field, and the breakdown voltage.
1. Capacitance: The capacitance of a conductor depends on its geometry and the dielectric material between the conductor and its surroundings. In the case of a sphere, the capacitance (C) can be calculated using the formula:
C = 4πε₀r
where r is the radius of the sphere and ε₀ is the permittivity of free space (8.85 x 10^-12 C²/(N.m²)). Plugging in the values, we get:
C = 4π(8.85 x 10^-12)(0.01) = 1.114 x 10^-10 F
Coulombs (C) is the unit of capacitance, so this capacitance value is in line with the unit of charge (Coulomb).
2. Electric field: The electric field (E) between the charged sphere and its surroundings directly affects the ability to hold a charge. The electric field can be calculated using the formula:
E = Q / (4πε₀r²)
where Q is the charge on the sphere. In this case, Q = 1 C and r = 0.01 m. Plugging in these values and solving the equation, we get:
E = (1) / (4π(8.85 x 10^-12)(0.01²)) ≈ 89878 N/C
This is a relatively high electric field.
3. Breakdown voltage: The breakdown voltage is the electric field strength at which the insulating material surrounding the conductor breaks down and creates a conductive path. For air, the breakdown voltage is approximately 3 x 10^6 V/m. As we calculated an electric field of 89878 N/C, this remains well below the breakdown voltage for air.
Therefore, based on these calculations and assumptions, a metal sphere of radius 1 cm can hold a charge of 1 Coulomb without any issue, as long as the assumptions hold and there are no specific constraints or limitations specified.
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