a metal ball of radius R is placed concentrically inside a hollow metal sphere of inner radius 2R & outer radius 3R. The ballis given a charge +2Q and the hollow sphere a total charge -Q. Calculate. electrostatic potential energy of system.???
To calculate the electrostatic potential energy of the system, we need to consider the contributions from both the metal ball and the hollow sphere.
The potential energy of the metal ball can be calculated using the formula:
U(ball) = (k * |Q1 * Q2|) / r,
where k is the electrostatic constant (k ≈ 9 * 10^9 Nm^2/C^2), Q1 and Q2 are the charges, and r is the distance between the charges.
In this case, the metal ball is positively charged with +2Q, and the hollow sphere is negatively charged with -Q. Since the charges have opposite signs, we use the absolute values for the charges when calculating potential energy.
Since the metal ball is placed concentrically inside the hollow sphere, the distance between the centers of the charges is the same as the radius of the ball, R.
U(ball) = (k * |2Q * Q|) / R.
The potential energy of the hollow sphere can be calculated using the formula:
U(sphere) = (k * |Q1 * Q2|) / r,
where Q1 and Q2 are the charges, and r is the distance between the charges.
In this case, the hollow sphere has a total charge of -Q, and the metal ball is the other charged object. The distance between the centers of the charges is the difference between the outer and inner radii of the sphere.
U(sphere) = (k * |-Q * 2Q|) / (3R - 2R) = (k * |2Q^2|) / R.
Now, to calculate the total electrostatic potential energy of the system, we add the potential energies of the metal ball and the hollow sphere:
U(total) = U(ball) + U(sphere)
= (k * |2Q * Q|) / R + (k * |2Q^2|) / R
= (k * |2Q^2 + 2Q^2|) / R
= (k * |4Q^2|) / R
= (4kQ^2) / R.
Therefore, the electrostatic potential energy of the system is (4kQ^2) / R.
To calculate the electrostatic potential energy of the system, we need to consider the interaction between the charged ball and the charged hollow sphere.
The potential energy (U) of a point charge (Q1) in the presence of an electric field (E) created by another charge (Q2) is given by the formula:
U = Q1 * V
Where V is the electric potential at the position of Q1 due to Q2.
In this case, the charged ball is inside the hollow sphere, and the inner surface of the hollow sphere creates an electric potential at the position of the ball.
Let's break down the problem step by step:
Step 1: Calculate the electric potential at the position of the ball.
Since the hollow sphere has a total charge of -Q, the electric field inside the hollow sphere (E_inner) can be zero as the charges on the inner surface cancel each other.
Therefore, the electric field just outside the ball and just inside the hollow sphere (E_outer) is given by:
E_outer = k * Q_inner_sphere / r_outer^2
Where k is the Coulomb's constant, Q_inner_sphere is the charge of the hollow sphere enclosed by the ball, and r_outer is the outer radius of the hollow sphere.
Since the charge enclosed by the ball is -Q (the total charge of the hollow sphere), we have:
E_outer = k * (-Q) / r_outer^2
Step 2: Calculate the electric potential at the position of the ball.
The electric potential (V_outer) at the position of the ball due to the electric field just outside the ball can be calculated using the formula:
V_outer = E_outer * r_outer
Substituting the value of E_outer, we get:
V_outer = k * (-Q) * r_outer / r_outer^2
Step 3: Calculate the electrostatic potential energy.
The electrostatic potential energy (U) of the charged ball due to the electric potential created by the charged hollow sphere can be calculated using the formula:
U = Q_ball * V_outer
Substituting the value of Q_ball (2Q) and V_outer, we get:
U = (2Q) * [k * (-Q) * r_outer / r_outer^2]
Simplifying the expression, we get:
U = - 2kQ^2 / r_outer
Therefore, the electrostatic potential energy of the system is - 2kQ^2 / r_outer.