Resultant of Different Permittivity substances

Ch4)In summary, the force experienced by a charge due to the field of the other charge can be calculated by solving Maxwell's Equations and finding the electric field due to the charges. This can be done by solving for the scalar potential in both regions and then stitching the solutions at the boundary. The problem is discussed in detail in Griffiths' "Introduction to Electrodynamics" and Jackson's "Classical Electrodynamics".
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Say the space is divided into two parts: one has permittivity e1 and other e2. A charge q1 is placed at a distance d1 from the interface of two media in region in permittivity e1 and another charge q2 is placed at a distance d2 in region of permittivity e2. What is the force experienced by a charge due to the field of the other charge? How to calculate it?
 
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Solve Maxwell's Equations, to find electric field due to charges. Finding the force is then trivial.

Electrostatics is fine for this problem, so (##\vec{E}## - electric field, ##\phi## - scalar potential, ##\rho## - charge density)

##\vec{\nabla}.\vec{E}=\rho/\epsilon##
##\vec{\nabla}\times\vec{E}=\vec{0}##

Which is satisfied if ##\vec{E}=-\vec{\nabla}\phi## and ##\epsilon\nabla^2\phi=-\rho##. Since you have point-charges, the charge density is zero almost everywhere, so begin by solving ##\nabla^2\phi=0## in both domains and then determine how to stitch the solution at the boundary.
 
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Problem (or something very similar) is treated in

Griffiths "Introduction to Electrodynamics", Example 4.8 (Ch4)
Jackson "Classical Electrodynamics", Section 4.4
 
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1. What is the significance of the resultant of different permittivity substances?

The resultant of different permittivity substances is the combined effect of the individual substances on the overall electric field in a given medium. It helps to understand how different materials affect the propagation of electromagnetic waves and can be used to design and optimize various electronic devices.

2. How is the resultant of different permittivity substances calculated?

The resultant of different permittivity substances is calculated by taking into account the individual permittivity values of each substance and their relative proportions in the medium. The resulting value is a vector quantity that represents the overall electric field strength in the medium.

3. Can the resultant of different permittivity substances be negative?

Yes, the resultant of different permittivity substances can be negative. This indicates that the combined effect of the substances results in a decrease in the overall electric field strength in the medium. This can occur when substances with high permittivity cancel out the effects of substances with low permittivity.

4. How does temperature affect the resultant of different permittivity substances?

Temperature can have a significant impact on the resultant of different permittivity substances. As the temperature of a substance increases, its permittivity may change, which can alter the overall resultant value. This is particularly important in materials used for electronic devices, as changes in temperature can affect their performance.

5. Can the resultant of different permittivity substances be used to determine the dielectric constant of a material?

Yes, the resultant of different permittivity substances can be used to determine the dielectric constant of a material. The dielectric constant is a measure of a material's ability to store electrical energy, and it can be calculated by dividing the resultant permittivity value by the permittivity of free space. This value is useful in understanding the behavior of materials in electric fields and designing electronic devices.

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