24.6
Calculations of Electric Potential I
Consider a ring of radius R with a uniform charge density λ. What will the electric potential be at point M, which is located on the axis of the ring…
Consider a spherical conductor with a radius of 20 centimeters and a charge of 40 micro-coulombs. What would be the electric potential at radial distances of 60 and 10 centimeters?
According to Gauss's law, the electric field inside the sphere is zero. So, to move a test charge within the sphere no work is done. The electric potential is constant inside the sphere and its value is the same as the potential at the surface.
Outside the sphere, the electric field follows the inverse square law. The electric potential is the integral of the dot product of the electric field and the displacement. At an infinite distance, the potential is defined as zero. This gives potential which varies inversely with the radial distance.
So, the electric potential is continuous, but the electric field is discontinuous.
Here, the known quantities are the sphere's radius, epsilon zero, and the charge. The unknown quantities are the electric potential at the two distances. By substituting the known quantities, the potential at 60 centimeters and at 10 centimeters are calculated.
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Q1: Why is electric potential constant inside a conductor?
According to Gauss's law, the electric field inside a conductor is zero. Since no work is done moving a test charge within the conductor, the electric potential remains constant throughout the interior and equals the potential at the surface. This uniform potential is a key property of conductors in electrostatic equilibrium.
Q2: How does electric potential vary with distance outside a charged sphere?
Outside a charged sphere, the electric field follows the inverse square law. The electric potential is calculated as the integral of the electric field and displacement, with potential defined as zero at infinite distance. This results in potential that varies inversely with radial distance from the sphere's center.
Q3: What is the relationship between electric field and electric potential continuity?
Electric potential is continuous across all regions, including at conductor boundaries. However, the electric field is discontinuous at these boundaries. This discontinuity arises because the electric field depends on the spatial rate of change of potential, which changes abruptly at conductor surfaces.
Q4: How is electric potential calculated for a uniformly charged ring?
For a charged ring, the ring is divided into infinitesimal arc elements. Since all elements are equidistant from a point on the ring's axis, the total potential equals the total charge divided by the distance from that point. Integration over all arc elements yields a result equivalent to placing all charge at a common distance.
Q5: What coordinate system is used to calculate potential at a point on a ring's axis?
The cylindrical coordinate system is used to calculate electric potential at a point on the axis of a charged ring. This system naturally describes the geometry of the ring and simplifies the integration process. Each arc element is characterized by its angular position and arc length.
Q6: How do you calculate electric potential at different radial distances from a sphere?
To find potential at specific distances, substitute the known quantities—sphere radius, permittivity of free space, and total charge—into the potential equation. For distances inside the sphere, potential equals the surface value. For distances outside, use the inverse square relationship to determine potential at each radial distance.
Q7: Why is the potential at a ring's axis point equal to total charge divided by distance?
Point M on the ring's axis is equidistant from all infinitesimal ring elements. Since potential is a scalar quantity, contributions from all elements add directly without vector complications. The total potential therefore simplifies to the total charge divided by the common distance, as if all charge were concentrated at one point.