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Chapter 6: Problem 33

The functions \(V_{1}(\rho, \phi, z)\) and \(V_{2}(\rho, \phi, z)\) both satisfy Laplace's equation in the region \(a<\rho

Short Answer

Expert verified
Based on the analysis and solutions provided: 1. The function \(V_1 + V_2\) satisfies Laplace's equation, but it is not guaranteed to satisfy the boundary conditions or be identical to \(V_1\). 2. The function \(V_1 - V_2\) satisfies Laplace's equation, but it is not guaranteed to satisfy the boundary conditions or be identical to \(V_1\). 3. The function \(V_1 + 3\) satisfies Laplace's equation but does not satisfy the boundary conditions and is not identical to \(V_1\). 4. The function \(V_1V_2\) does not satisfy Laplace's equation, the boundary conditions, or be identical to \(V_1\). Thus, none of the functions are guaranteed to satisfy Laplace's equation, the boundary conditions, and be identical to \(V_1\) under the given conditions.

Step by step solution

01

Check if the functions satisfy Laplace's equation

We have the functions \(V_1 (\rho, \phi, z)\) and \(V_2 (\rho, \phi, z)\), both satisfying Laplace's equation in the specified region. So, let's check if the following functions also satisfy Laplace's equation: 1. \(V_1 + V_2\): Since Laplace's equation is a linear partial differential equation, their sum also satisfies Laplace's equation. 2. \(V_1 - V_2\): The difference between two solutions to Laplace's equation is also a solution to Laplace's equation. 3. \(V_1 + 3\): Adding a constant to the solution of Laplace's equation also satisfies Laplace's equation. 4. \(V_1 V_2\): The product of two solutions to Laplace's equation is not a solution to Laplace's equation, in general.
02

Check if the functions satisfy potential values at boundary surfaces

Now let's check if the following functions satisfy the given potential values at boundary surfaces: 1. \(V_1 + V_2\): The sum will satisfy the boundary conditions if both \(V_1(\rho=a)\) and \(V_2(\rho=a)\) are equal to 100 V. Otherwise, it will not. 2. \(V_1 - V_2\): The difference will satisfy the boundary conditions if \(V_1(\rho=a)\) and \(V_2(\rho=a)\) are equal to 100 V. Otherwise, it will not. 3. \(V_1 + 3\): Since \(V_1(\rho=a)\) is equal to 100 V, adding 3 to it does not satisfy the boundary condition. 4. \(V_1V_2\): The product does not satisfy the boundary conditions, in general, unless a specific relationship between \(V_1\) and \(V_2\) exists that causes their product to equal 100 V at \(\rho=a\).
03

Check if the functions are identical to \(V_1\)

Finally, let's check if the functions are identical to \(V_1(\rho, \phi, z)\): 1. \(V_1 + V_2\): The sum will not be identical to \(V_1\) unless \(V_2\) is zero everywhere, which is not given in the problem statement. 2. \(V_1 - V_2\): The difference will not be identical to \(V_1\) unless \(V_2\) is zero everywhere, which is not given in the problem statement. 3. \(V_1 + 3\): Adding a constant does not make a function identical to its original version. Thus, this is not identical to \(V_1\). 4. \(V_1 V_2\): The product of two functions is not identical to one of the original functions, in general. This is not identical to \(V_1\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Electromagnetic Theory and Laplace's Equation
Electromagnetic theory is a branch of physics that deals with the study of electromagnetic forces, a type of physical interaction that occurs between electrically charged particles. One of the fundamental equations in electromagnetic theory is Laplace's equation. This equation is used to describe the behavior of electric potentials in a field where there are no free charges present, under the assumption of static conditions.

Laplace's equation is mathematically represented as abla^{2}V = 0, where abla^{2}is the Laplacian operator and V is the electric potential. The equation is pivotal in determining the potential function in a given region when the boundary conditions are known. It's essential in many areas, including electrostatics, fluid mechanics, and heat conduction. Moreover, in the context of electromagnetic theory, solving Laplace's equation with specific boundary conditions is crucial to finding the electric field distribution generated by a static charge distribution.
Understanding Partial Differential Equations
Partial differential equations (PDEs) are equations involving the partial derivatives of a function of multiple variables. They serve to model a wide range of phenomena in engineering, physics, and other sciences. Laplace's equation is a second-order linear PDE.

To solve PDEs like Laplace's equation, it's necessary to consider boundary conditions which provide the values of the function or its derivatives at the boundaries of the domain of interest. The solution to a PDE typically involves integration, but the process can be complex depending on the geometry of the domain and the nature of the conditions imposed. It's fascinating to note that PDEs can take on infinite solutions, and boundary conditions help in pinpointing the unique solution relevant to the physical problem.
The Role of Boundary Conditions
Boundary conditions are an integral part of solving equations in physics and engineering, as they specify the solution behavior at the boundaries of the domain. In the case of Laplace's equation, the boundary conditions determine the values of the electric potential at the domain’s edges. There are different types of boundary conditions: Dirichlet, where the potential is defined on the boundary; Neumann, where the derivative of the potential is given; and mixed, involving both potential and its derivative.

In the step-by-step solutions described, different boundary conditions are checked for various manipulations of the functions V_1 and V_2. This is a crucial step in the process, as even if a particular combination (e.g., V_1 + V_2) satisfies Laplace's equation, it may not necessarily satisfy the specified boundary conditions.
Electric Potential in Electromagnetic Theory
Electric potential, often symbolized as V, is a scalar quantity that represents the electric potential energy per unit charge at a point in space. In electromagnetic theory, electric potential is used to describe the potential energy landscape for charges within an electric field. Its value is directly related to the work done by an external force to bring a unit positive charge from infinity to that point.

The concept of electric potential is particularly significant when it comes to understanding concepts such as voltage, capacitance, and energy storage in electric fields. It's also crucial in solving Laplace's equation, where the potential values at the boundaries play a key role in defining the solution. In the exercise provided, the boundaries are set to specific voltages, creating a well-defined problem for finding the electric potential within the given region.

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Most popular questions from this chapter

Given the potential field \(V=\left(A \rho^{4}+B \rho^{-4}\right) \sin 4 \phi:(a)\) Show that \(\nabla^{2} V=0\). (b) Select \(A\) and \(B\) so that \(V=100 \mathrm{~V}\) and \(|\mathbf{E}|=500 \mathrm{~V} / \mathrm{m}\) at \(P(\rho=1\), \(\left.\phi=22.5^{\circ}, z=2\right) .\)

Construct a curvilinear-square map of the potential field between two parallel circular cylinders, one of \(4 \mathrm{~cm}\) radius inside another of \(8 \mathrm{~cm}\) radius. The two axes are displaced by \(2.5 \mathrm{~cm}\). These dimensions are suitable for the drawing. As a check on the accuracy, compute the capacitance per meter from the sketch and from the exact expression: $$C=\frac{2 \pi \epsilon}{\cosh ^{-1}\left[\left(a^{2}+b^{2}-D^{2}\right) /(2 a b)\right]}$$ where \(a\) and \(b\) are the conductor radii and \(D\) is the axis separation.

Let \(V=(\cos 2 \phi) / \rho\) in free space. (a) Find the volume charge density at point \(A\left(0.5,60^{\circ}, 1\right) .(b)\) Find the surface charge density on a conductor surface passing through the point \(B\left(2,30^{\circ}, 1\right)\).

Two coaxial conducting cylinders of radius \(2 \mathrm{~cm}\) and \(4 \mathrm{~cm}\) have a length of \(1 \mathrm{~m}\). The region between the cylinders contains a layer of dielectric from \(\rho=c\) to \(\rho=d\) with \(\epsilon_{r}=4\). Find the capacitance if \((\) a) \(c=2 \mathrm{~cm}, d=3 \mathrm{~cm} ;\) (b) \(d=4 \mathrm{~cm}\), and the volume of the dielectric is the same as in part \((a)\).

The derivation of Laplace's and Poisson's equations assumed constant permittivity, but there are cases of spatially varying permittivity in which the equations will still apply. Consider the vector identity, \(\nabla \cdot(\psi \mathbf{G})=\mathbf{G} \cdot \nabla \psi+\) \(\psi \nabla \cdot \mathbf{G}\), where \(\psi\) and \(\mathbf{G}\) are scalar and vector functions, respectively. Determine a general rule on the allowed directions in which \(\epsilon\) may vary with respect to the local electric field.
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