What is the Image method?

The method of images (or mirror method) is a mathematical tool used to solve different scales, in which the required task domain is expanded by adding mirror images related to the symmetric hyperplane. As a result, certain boundary conditions are automatically satisfied with the presence of a mirror image, making it easier to solve the problem.

Working principle of the image method

The method of images works because the solution of Laplace's calculated value specified in a closed area is different from a solution for Poisson's calculation with a specified value in a given closed area and a specified charging congestion within a closed circuit. Therefore, if the goal is to gain power within a particular region, it can be done by guessing the work that satisfies Poisson's figure of given charge congestion and a fair border value. Once this is done, the alternative theory mentioned in the previous paragraph confirms that it should be the right solution.

Once a user calculate the potential above the plane using the image charge, one should find that the user gets the correct value on the plane itself (zero), the correct value at infinity above the plane (zero), and the correct charge density above the plane (i.e., −ϵ₀ times the Laplacian), that is, zero everywhere except at the location of the point charge, where superposition guarantees that it is still the correct delta function. The uniqueness theorem applies to the entire volume above the plane, which is bounded by the plane itself and the surface at infinity above the plane. Therefore, the potential in this area calculated by the image method is correct.

Method of image charge

The image charging method is utilized in electrostatics to easily calculate or visualize the charge distribution of the charging field inside the vicinity of the operating area. It is primarily based on the fact that the tangential part of the electric field at the surface of the conductor is zero, and that the electric field E in a selected area is defined differently by its normal part over the area that connects this circuit (variance time).

Uniqueness and existence theorem

Uniqueness theorem

In mathematics, the uniqueness theorem is a variant that assumes that certain conditions are met, or that all things that meet the conditions are equal.

A uniqueness theorem, also called a unicity theorem, refers to a uniqueness of a mathematical object, usually meaning that only one object fulfills the condition of a given property, or that all objects in a given category are equal (i.e., can be represented equality model). This uniqueness theorem usually means that an object is individually determined by a specific set of data. A unique word is sometimes replaced by a essentially unique, whenever one wants to emphasize that the uniqueness refers to the sub-structure, while the form can be different in every way that does not affect the mathematical content.

Existence theorem

Existence theorems are usually easy to identify because they say that there is at least one thing with the necessary structure. This type of theory is usually confirmed in one of two ways:

• If possible, use an algorithm (process) to construct at least one object with the required properties.
• Sometimes, especially in highly developed mathematical theories, a clear construction is not possible. Therefore, one should be able to find a common argument that confirms the existence of an imaginary object, without being able to give a real example of it.

Uses of image method

• The imaging method can also be used in magnetostatics to calculate the magnetic field that is close to the superconducting factor.
• In electrostatic boundary value problem, the problems are solved by replacing boundary surfaces with appropriate image charges without the equations. This helps us as solving the electrostatic boundary value problems are very difficult using Laplace and Poisson equations.

Common Mistakes

The photographic method works best when the border is flat and the spread is centered on the geometry. This allows for easy mirror-like display distribution to satisfy a wide variety of border conditions.

Context and Applications

In each of the expert exams for undergraduate and graduate publications, this topic is huge and is mainly used for:

• Bachelor of technology in the electrical and electronic department
• Bachelor of Science in physics
• Master of Science in physics

Practice Problems

Q1. Calculate the electric field intensity of a line charge of length 2m and potential 24V.

1. 24
2. 12
3. 0.083
4. 12.67

Correct option: (b)

Explanation: Power field is given a measure of power and distance or length.

$$\begin{gathered} E=\frac{V}{d} \\ E=\frac{24}{2} \\ E=12 \mathrm{V}/\mathrm{m} \end{gathered}$$

Q2. Find the dissipation factor when series resistance is 5 ohm and capacitive resistance is 10 units.

1. 2
2. 0.5
3. 1
4. 0

Correct option: (b)

Explanation: The dissipation factor is nothing but the tangent of loss angle of loss tangent.

$$\begin{gathered} \tan \delta =\frac{\mathrm{Series} \mathrm{resistance}}{\mathrm{Capacitive} \mathrm{resistance}} \\ \tan \delta =\frac{5}{10} \\ \tan \delta =0.5 \end{gathered}$$

Q3. Find the energy stored by the capacitor 3F having a potential of 12V across it.

1. 432
2. 108
3. 216
4. 54

Correct option: (c)

Explanation: The energy stored in a capacitor is given by,

$$\begin{gathered} E=0.5C{V}^2 \\ E=0.5\left(3\right){\left(12\right)}^2 \\ E=0.5\left(432\right) \\ E=216 \mathrm{units} \end{gathered}$$

Q4. By method of images, the problem can be easily calculated by replacing the boundary with which polygon?

1. Rectangle
2. Trapezoid
3. Square
4. Triangle

Correct option: (d)

Explanation: If any field or power needs to be calculated by line charging or a solid cable or a fixed cylinder, the imaging method uses a triangle that converts a dimensional problem into a one-sided analysis. From this, the result can be calculated.

Q5. A material with zero resistivity is said to have __________.

1. Zero Conductance
2. Infinite conductance
3. Zero resistance
4. Infinite resistance

Correct option: (c)

Explanation: Since the resistivity is directly proportional to the resistance when the resistivity is zero, the resistance is also zero. So, one gets zero resistance. The infinite conductance option is also possible, but not possible. As there is always some loss in the form of heat, so continuous operation is not possible, but a short circuit (zero resistance) is possible.