A metal sphere of radius R carries a total charge Q . What is the force of repulsion between the

A metal sphere of radius R carries a total charge Q . What...

Key Concepts

Electrostatics

Electrostatics involves the study of electric charges at rest and the forces and fields they generate. In this context, understanding the mutual forces between parts of a charged object is fundamental, as it requires analyzing how like charges repel one another and how electric fields are established in and around conductors.

Surface Charge Distribution

For conductors, excess charge resides on their surface and rearranges itself to maintain equilibrium. In a spherical conductor, the charge distributes uniformly over the surface, leading to a radial electric field. This uniform surface charge distribution is crucial for determining the resulting forces on different parts of the conductor.

Electrostatic Pressure

Electrostatic pressure refers to the mechanical pressure exerted on the surface of a charged conductor due to the repulsive forces between like charges. This concept allows one to relate the electric field strength at the surface with the force per unit area, providing a way to compute forces between different regions on the surface.

Maxwell Stress Tensor

The Maxwell stress tensor is a mathematical tool used in electromagnetism to calculate the force and torque experienced by a system due to electric and magnetic fields. In problems where forces are distributed over surfaces, like in the case of a charged sphere, it provides a systematic way to evaluate the net force acting between different parts of the charge distribution.

Coulomb's Law Integration

Coulomb's law governs the force between two point charges, and in extended distributions, the total force can be determined by integrating the contributions from infinitesimal charge elements. This method is often used to calculate net forces in systems with continuous charge distributions by summing the interactions between all pairs of differential charges.

Transcript

0:00 Hi there.

00:01 So for this problem, we are told that a metal sphere of radius capital r carries a total charge q, as is shown here in this figure.

00:11 Now, what is the force of repulsion between the northern hemisphere and the southern hemisphere? in here, we have the drawing of the northern hemisphere.

00:26 So we note that inside, it's right in here, inside this, the electric field is equal to zero.

00:38 However, upside, the electric field corresponds to the electric field of a point charge that we know is 1 over 4 times pi times epsilon sub 0 times a charge, capital q divided by the radius square, this in the radial direction.

01:00 So the average electric field is equal to 1 over 2 times 1 over 4 times pi times epsilon sub 0 times the charge capital q divided by the radius capital r to the square, this in the radial direction.

01:24 So if we want to calculate the force per unit length, we're going to call this in the c.

01:31 Direction, that is going to be the charge density, the charge surface density times the average electric field in the c direction.

01:46 Now in this case, we know that the surface charge density is equal to the charge q divided by the surface area that we know is 4 times pi times the radius r to the square.

02:00 So with this, if we want to calculate the force in the cid direction, that is going to be the integral of the force per unit length times the differential in area.

02:13 So we substitute of this expression, so we will have the charge capital q divided by four times pi times the radius r, capital r to the square, this times 1 over 2.

02:27 And this times 1 over 4 times pi xepsilon sub 0 times the charge q divided by the radius capital r to the square.

02:37 And this cosine of theta times the radius capital r square...

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