Gauss' law -- Conductor with a cavity

In summary, a point charge of -3 ##\mu##C is distributed on the surface of a conductor when the electric field is zero inside the conductor.
  • #1
Physicslearner500039
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6
Homework Statement
An isolated conductor has net charge +10 X 10 -6 C and a cavity with a point charge q = +3.0 X 10-6 C. What is the charge on (a) the cavity wall and (b) the outer surface?
Relevant Equations
NA
P21.PNG

I really don't understand the theory of the above kind of questions. But from the little theory i understand the Electric field is 0 inside the conductor and all the charge goes to the surface and distributes equally.

a. Since the E=0 inside the conductor the point charge distributes outside the cavity to make the net charge as 0. Hence the cavity develops a charge of -3uC on its surface. I am not 100% sure how it works and the negative sign. Please advise.

b. If i apply the Gauss Law with a surface around the conductor the net charge enclosed is 10uC + 3uC = 13uC distributes on the surface of the conductor.
 
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  • #2
For (a) think about Gauss's law just inside the conductor at its inner surface.
 
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  • #3
Physicslearner500039 said:
  1. Electric field is 0 inside the conductor ##\qquad## correct
  2. and all the charge goes to the surface ##\qquad## correct (surfaces)
  3. and distributes equally ##\qquad## ##\qquad## ##\qquad\quad## not correct
ad 2: there can be more surfaces, like in this exercise
ad 3: equally distributed happens in specific cases (with some symmetry). The basic rule is that the electric field lines are perpendicular to the surface(s)
(if not, there would be a component along the surface, and the charge would re-distribute until that component is zero everywhere).

In your exercise it is not given that the cavity is in the centre of the sphere, nor that the charge in in the centre of the cavity. But you can, as @hutchphd indicates, make good use of Gauss' theorem.
 
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  • #4
S21.PNG

I have drawn a diagram to understand, the q1 = 3uC. If we consider the gaussian surface the Flux is zero
∅ = 0; ∈0*∅ = q_enclosed;
q_enclosed = 0;
Let the charge on the cavity is q_cavity;
q_cavity + q1 = 0;
q_cavity = -3uC
This i understood. The confusion is for the net charge let us say q_net.
The same principle i apply ∅ = 0;
The charge on the surface is q_surface;
q_surface + qnet +q_cavity+q1 = 0;
q_surface = -qnet; = -10uC; The answer is wrong and i am completely wrong, where i am making the mistake. Please advise.
 
  • #5
Physicslearner500039 said:
q_cavity = -3uC
Correct.
The conductor charge is +10 ##\mu##C.
What is left of the conductor charge is sitting on the outside of the conductor. How much is that if you subtract the -3 ##\mu##C from the original +10 ##\mu##C ?
 

Related to Gauss' law -- Conductor with a cavity

What is Gauss' law?

Gauss' law is a fundamental law in electromagnetism that relates the electric flux through a closed surface to the charge enclosed by that surface.

What is a conductor with a cavity?

A conductor with a cavity is a conducting material with a hollow space inside it. This can be thought of as a metal box with an empty interior.

How does Gauss' law apply to a conductor with a cavity?

In a conductor with a cavity, the electric field inside the cavity is zero. This is because any excess charge on the surface of the cavity will redistribute itself in a way that cancels out the electric field inside.

What is the electric field outside of a conductor with a cavity?

The electric field outside of a conductor with a cavity is the same as that of a conductor without a cavity. It is perpendicular to the surface of the conductor and its strength decreases as you move away from the surface.

How is the electric field affected by the shape and size of the cavity in a conductor?

The electric field outside of a conductor with a cavity is affected by the shape and size of the cavity. For example, a larger cavity will result in a weaker electric field, while a smaller cavity will result in a stronger electric field. The shape of the cavity also plays a role in determining the strength and direction of the electric field.

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