Induced charge in image problem

In summary, the conversation is about finding the induced charge density in the classic image problem. The person asking for help has tried to use polar coordinates but believes they made a mistake. The other person points out that the dipole approximation cannot be used in this problem because the potential does not go to zero on the plane. The first person disagrees, saying that their potential does go to zero on the plane. The second person explains that the potential should be expressed as the sum of potentials from the point charge and its image, and suggests using cylindrical coordinates.
  • #1
Kaguro
221
57

Homework Statement


Find induced charge density in the classic image problem.

Sorry, I uploaded my work in form of an image instead of typing.

I have tried to use polar coordinates to solve this. But i must have done something wrong. Please help me find the mistake.

15512082946811688506627.jpg
 

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  • #2
I do not believe you can use the dipole approximation here. It is valid for ##r>>d##. Your potential does not go to zero on the plane which is the boundary condition in this problem.
 
  • #3
kuruman said:
I do not believe you can use the dipole approximation here. It is valid for ##r>>d##. Your potential does not go to zero on the plane which is the boundary condition in this problem.

But my potential does go to 0 on the plane as theta is 90 then and so cos theta is 0.
 
  • #4
The entire plane is an equipotential therefore the expression must evaluate to zero for any value of angle ##\theta## and must also be zero at the origin. You should express the potential as the sum of the potentials from the point charge and its image. You would be wise to use cylindrical coordinates.
 

What is induced charge in image problem?

Induced charge in image problem is a concept in electrostatics that describes the redistribution of charges on a conductor due to the presence of another charged object nearby. This redistribution is caused by the electric field of the charged object and can result in a net charge on the conductor, known as the induced charge.

How is induced charge related to the concept of image charges?

Induced charge is closely related to the concept of image charges, which describes the imaginary charges that are created on a conductor when it is placed in the electric field of another charged object. These image charges are used to simplify the calculations of the electric potential and electric field in complex electrostatic problems.

What is the importance of studying induced charge in image problem?

Studying induced charge in image problem is important because it allows us to understand and analyze the behavior of electric fields and charges in complex systems. It also has practical applications in various fields such as electronics, electromagnetism, and telecommunications.

Can induced charge be both positive and negative?

Yes, induced charge can be both positive and negative. The sign of the induced charge depends on the direction of the electric field of the charged object and the properties of the conductor, such as its shape and material.

How can we calculate the induced charge on a conductor?

The induced charge on a conductor can be calculated using the principle of charge conservation, which states that the total charge on the conductor must remain constant. This can be done by considering the electric field and potential created by the charged object and solving the resulting equations.

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