A metal sphere with radius R1 has a charge Q1 . Take the...

A metal sphere with radius $R_1$ has a charge $Q_1$ . Take the electric potential to be zero at an infinite distance from the sphere. (a) What are the electric field and electric potential at the surface of the sphere? This sphere is now connected by a long, thin conducting wire to another sphere of radius $R_2$ that is several meters from the first sphere. Before the connection is made, this second sphere is uncharged. After electrostatic equilibrium has been reached, what are (b) the total charge on each sphere; (c) the electric potential at the surface of each sphere; (d) the electric field at the surface of each sphere? Assume that the amount of charge on the wire is much less than the charge on each sphere.

Key Concepts

Gauss’s Law and Spherical Symmetry

Gauss’s Law allows the computation of the electric field produced by a symmetric charge distribution by considering the flux through a spherical surface. For a charged sphere, the field outside behaves as if all the charge were concentrated at its center. This symmetry simplifies the evaluation of both the electric field and the electric potential, since the field at a distance r from the center only depends on the total enclosed charge.

Electric Potential of a Charged Sphere

The electric potential at any point outside a charged sphere is equivalent to that of a point charge located at its center, given by V = kQ/r where k is Coulomb’s constant. At the surface of the sphere, the potential is V = kQ/R, where R is the radius of the sphere. This concept is crucial for calculating potentials when dealing with isolated charged spheres and serves as a starting point in more complex configurations.

Electrostatic Equilibrium in Connected Conductors

When two or more conductors are connected by a conducting wire, they come to the same electric potential in electrostatic equilibrium. This principle implies that any difference in potential initially present will drive a redistribution of charge until uniform potential is achieved. Although the geometry of each conductor affects the local electric field, the final state is determined by the condition of equal potential throughout the system.

Conservation of Charge

The conservation of charge principle states that the total charge in an isolated system remains constant. When two conductors, such as metallic spheres, are connected, charge may transfer between them until equilibrium is reached, but the sum of their charges remains the same. This concept is essential for determining the final charge distribution on each sphere when they are electrically connected.

Surface Electric Field of Conductors

For conductors in electrostatic equilibrium, the electric field just outside the surface is perpendicular to the surface and is directly related to the surface charge density. At the surface of a charged sphere the magnitude of the electric field is given by E = kQ/R^2. This relationship is used to connect the macroscopic charge on the conductor with the local electric field, and it plays an important role when analyzing the behavior of connected conducting bodies.

Transcript

00:01 Let's talk about this question.

00:02 We are given that a metal sphere radius r1 has a charge of q1, and we have to take the net -elected potential to be zero at infinite distance from the sphere.

00:10 Part a talks about what is the electric field and the electric potential of the surface of the sphere? so if this is a sphere having a charge of q1, and this is a metal sphere, so definitely everything will be on the surface.

00:23 That's the property of the metal.

00:25 If we talk about the electric potential or the electric field at anywhere on the surface, then this charge actually, a sphere actually behaves like a point charge at a distance of r1 from that point.

00:39 So we know that if we talk about the electric potential, that's going to be q1 over 4 pi aps knot r1.

00:46 This can be written as kq1 over r1.

00:50 So this is the required potential.

00:53 And if we talk about the electric field, that's going to be q1 over 4 pi aps knot r1 square.

01:01 So that's going to be, can be written as kq1 over r1 square.

01:06 So that is the required electric potential.

01:09 There's a required of, there's a required electric field and electric potential at the surface.

01:15 Now this sphere, the sphere is now connected by a long, thin conducting wire to another sphere of radius r2.

01:22 There are several meters from the first sphere.

01:24 So there's a reason why they are seeing that it is several meters from the first sphere.

01:28 I will talk about it.

01:31 Before the connection is made, the second sphere is uncharged.

01:34 So let's see how this looks like.

01:36 So we have this as the first sphere with q1 and r1.

01:43 And this is the thin cable with which we are connecting another sphere of some unknown charge q and radius r2.

01:52 Now, after the electrostatic equilibrium has reached, what are the total charges on each sphere? all right.

02:00 So, since they are connected with the conducting a wire, it means that the charge will flow from q1 to the other uncharged sphere q.

02:13 This is uncharged.

02:14 I'm really sorry.

02:15 There's no q here.

02:16 Q such that their both potential remains the same.

02:20 Now, the point is over to q1, the potential is definitely due to q1 and due to this sphere.

02:28 But since they have said that that is several meters from the first sphere, which means that we can ignore the effect of this charge over here and vice versa of this charge over here.

02:42 So at this surface, the only potential is created by the charge q1 and the same is applicable for.

02:50 Rather sphere as well.

02:51 So let's say the x charge moves from here to here...

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