Two point charges are enclosed by a spherical conducting...

Two point charges are enclosed by a spherical conducting shell that has an inner and outer radius of $13.0$ and $15.2 \mathrm{~cm}$, respectively. One point charge has a charge of $\mathrm{q} 1=8.30 \mu \mathrm{C}, q_{1}=8.30 \mu \mathrm{C}$, while the second point charge has an unknown charge $q_{2}$. The conducting shell is known to have a net electric charge of $-2.40 \mu \mathrm{C}$, but measurements find that the charge on the outer shell is $+3.70 \mu \mathrm{C}$. Determine the charge $q_{2}$ of the second point charge. Example $\underline{16-8}$

Key Concepts

Gauss's Law

Gauss's Law is a fundamental principle of electromagnetism that relates the net electric flux through a closed surface to the total charge enclosed within that surface. It is especially useful in problems with symmetrical geometries such as spherical conductors, where it allows one to calculate the electric field and deduce charge distributions based on symmetry arguments.

Conductors in Electrostatic Equilibrium

In electrostatic equilibrium, conductors exhibit the property that the electric field inside the conducting material is zero. This forces any excess charge to reside entirely on the surface of the conductor. When internal charges are present, the conductor redistributes surface charges in such a way as to cancel the internal electric fields, thereby maintaining equilibrium.

Induced Charge Redistribution

Induced charge redistribution refers to the process by which charges in a conductor rearrange themselves in response to external or internal charges. For instance, if a conductor encloses an internal charge, an equal and opposite charge is induced on its inner surface to neutralize the field inside the conductor, while any remaining net charge is found on the outer surface.

Charge Conservation

Charge conservation is the principle that the total electric charge in an isolated system remains constant despite the movement or rearrangement of charges. In the context of conductive shells, any net charge measured on the shell must be consistent with both the induced charge on the inner surface and any free charge on the outer surface, ensuring that all contributions add up to the initially present charges.

Transcript

00:01 In this problem we have a spherical shell that includes two point charges.

00:09 The first is this q1 that we know what is the value of this charge.

00:16 And the second, we don't know.

00:18 We actually want to compute the value of the second charge.

00:22 Also, the problem says that this shell is a conductor.

00:27 That means that the electric charge is located on the surface.

00:32 Of the shelf, so there will be an amount of electric charge that i will call capital q1 on this internal surface, and another on the external surface that i have called capital q2.

00:48 The problem says that we know the value of capital q2 is 3 .70 microtum, and also we know what is that total electric charge of the shell.

01:02 So we know the value of this capital q1 plus capital q2 has this value.

01:10 So from these two values here, we can get actually the value of capital q1, right? because we know that this is 3 .70 microcolum.

01:27 So from this equation we can get the value of capital q1.

01:32 And this is actually a minus 6 .10 microcolum.

01:40 Okay.

01:42 In order to compute this small q2, that is the other point, charge inside the shell, we are going to use the gauss law.

01:58 The gauss law says that the total electric flux or the flux of the electric field through a closed surface is equal to the total charge enclosed by the surface divided by the electric permittivity...

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