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Chapter 1: Problem 8

\(\varphi\) is a scalar satisfying Laplace's equation \(\nabla^{2} \varphi=0\). Show that \(\nabla \varphi\) is both solenoidal and irrotational.

Short Answer

Expert verified
\( \nabla \varphi \) is solenoidal because \( \nabla \cdot \nabla \varphi = 0 \) and irrotational because \( \nabla \times \nabla \varphi = 0 \).

Step by step solution

01

Understanding the Given Conditions

We are given that \(abla^2 \varphi = 0\), which is a scalar potential field satisfying Laplace's equation. Our goal is to show that \( abla \varphi \) is both solenoidal and irrotational. A vector field \( \mathbf{F} \) is solenoidal if \( abla \cdot \mathbf{F} = 0 \) and irrotational if \( abla \times \mathbf{F} = 0 \).
02

Showing Solenoidal Property

To show that \( abla \varphi \) is solenoidal, we need to calculate \( abla \cdot (abla \varphi) \). We know that this equals \( abla^2 \varphi \). Since \( abla^2 \varphi = 0 \) by assumption, it follows that \( abla \cdot (abla \varphi) = 0 \). This proves that \( abla \varphi \) is solenoidal.
03

Showing Irrotational Property

To demonstrate that \( abla \varphi \) is irrotational, we need to calculate \( abla \times (abla \varphi) \). It is a vector calculus identity that the curl of a gradient is always zero, i.e., \( abla \times (abla \varphi) = 0 \). Hence, \( abla \varphi \) is irrotational.
04

Conclusion

Since we have shown that \( abla \varphi \) is both solenoidal (\( abla \cdot abla \varphi = 0 \)) and irrotational (\( abla \times abla \varphi = 0 \)), we conclude that the conditions required by the problem are satisfied.

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Key Concepts

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

Scalar Potential Field
A scalar potential field is a field where each point in space is associated with a scalar value. This is often used in physics to describe potential energy landscapes, like gravitational or electrostatic fields.
In the context of Laplace's equation, the scalar potential field is denoted by the scalar function \( \varphi \). When \( \varphi \) satisfies Laplace's equation, \( abla^{2} \varphi = 0 \), this implies that \( \varphi \) is in a state of equilibrium, meaning there are no local variations that would create a net flow or change.
This equilibrium is crucial because it helps to show that any derivatives of \( \varphi \) form a well-behaved vector field that can have special properties like being solenoidal or irrotational.
Solenoidal Vector Field
A solenoidal vector field is one where the divergence of the field is zero everywhere. Mathematically, for a vector field \( \mathbf{F} \), being solenoidal means \( abla \cdot \mathbf{F} = 0 \).
This concept is fundamental in understanding fluid dynamics and electromagnetism where solenoidal fields often represent incompressible flows or magnetic fields.
For our scalar potential field \( \varphi \), by calculating \( abla \cdot (abla \varphi) \) and recognizing that this equals \( abla^2 \varphi \), which is already given as zero, we immediately conclude that the gradient \( abla \varphi \) is solenoidal.
This means there's no net 'source' or 'sink' anywhere in the vector field, maintaining a consistent flow.
Irrotational Vector Field
An irrotational vector field is described by the condition that its curl is zero throughout the field. For a vector field \( \mathbf{F} \), this is expressed as \( abla \times \mathbf{F} = 0 \).
Irrotational fields are often encountered in physics in the analysis of fluid flows and theories of gravitation and electrostatics.
The property of irrotationality suggests that locally, the vector field exhibits no rotation or swirling behavior.
In the context of our scalar field \( abla \varphi \), using the vector calculus identity \( abla \times (abla \varphi) = 0 \), we find that \( abla \varphi \) is irrotational.
Thus, \( abla \varphi \) as a vector field is free from rotational influence, emphasizing its smooth structure.
Vector Calculus Identities
Vector calculus identities are crucial tools used to simplify and understand complexities in multivariate problems like fluid dynamics or electromagnetism.
Key identities include the divergence and curl operations, which help us understand the behavior of vector fields.
Important identities relevant to this problem are:
  • The Divergence of a Gradient: \( abla \cdot (abla \varphi) = abla^2 \varphi \)
  • The Curl of a Gradient: \( abla \times (abla \varphi) = 0 \)
These rules help in verifying the conditions that a field is solenoidal or irrotational.
Having a solid grasp of these identities simplifies the analysis, allowing for understanding of fields in terms of sources, sinks, or rotational components.
Mastery of these concepts enables physicists and engineers to predict and manipulate natural phenomena efficiently.

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Most popular questions from this chapter

Given $$ \mathbf{a}^{\prime}=\frac{\mathbf{b} \times \mathbf{c}}{\mathbf{a} \cdot \mathbf{b} \times \mathbf{c}}, \quad \mathbf{b}^{\prime}=\frac{\mathbf{c} \times \mathbf{a}}{\mathbf{a} \cdot \mathbf{b} \times \mathbf{c}}, \quad \mathbf{c}^{\prime}=\frac{\mathbf{a} \times \mathbf{b}}{\mathbf{a} \cdot \mathbf{b} \times \mathbf{c}}, $$ and \(\mathbf{a} \cdot \mathbf{b} \times \mathbf{c} \neq 0\), show that (a) \(\mathbf{x}^{\prime} \cdot \mathbf{y}=\delta_{x y},(\mathbf{x}, \mathbf{y}=\mathbf{a}, \mathbf{b}, \mathbf{c})\) (b) \(\mathbf{a}^{\prime} \cdot \mathbf{b}^{\prime} \times \mathbf{c}^{\prime}=(\mathbf{a} \cdot \mathbf{b} \times \mathbf{c})^{-1}\), (c) \(\mathbf{a}=\frac{\mathbf{b}^{\prime} \times \mathbf{c}^{\prime}}{\mathbf{a}^{\prime} \cdot \mathbf{b}^{\prime} \times \mathbf{c}^{\prime}}\).

The usual problem in classical mechanics is to calculate the motion of a particle given the potential. For a uniform density \(\left(\rho_{0}\right)\), nonrotating massive sphere, Gauss's law of Section \(1.14\) leads to a gravitational force on a unit mass \(m_{0}\) at a point \(r_{0}\) produced by the attraction of the mass at \(r \leq r_{0}\). The mass at \(r>r_{0}\) contributes nothing to the force. (a) Show that \(\mathbf{F} / m_{0}=-\left(4 \pi G \rho_{0} / 3\right) \mathbf{r}, 0 \leq r \leq a\), where \(a\) is the radius of the sphere. (b) Find the corresponding gravitational potential, \(0 \leq r \leq a\). (c) Imagine a vertical hole running completely through the center of the earth and out to the far side. Neglecting the rotation of the earth and assuming a uniform density \(\rho_{0}=5.5 \mathrm{gm} / \mathrm{cm}^{3}\), calculate the nature of the motion of a particle dropped into the hole. What is its period? Note. \(\mathbf{F} \propto \mathbf{r}\) is actually a very poor approximation. Because of varying density, the approximation \(\mathbf{F}=\) const., along the outer half of a radial line and \(\mathbf{F} \propto \mathbf{r}\) along the inner half is a much closer approximation.

The origin of Cartesian coordinates is at the Earth's center. The moon is on the \(z\) -axis, a fixed distance \(R\) away (center-to-center distance). The tidal force exerted by the moon on a particle at the earth's surface (point \(x, y, z)\) is given by $$ F_{x}=-G M m \frac{x}{R^{3}}, \quad F_{y}=-G M m \frac{y}{R^{3}}, \quad F_{z}=+2 G M m \frac{z}{R^{3}} . $$ Find the potential that yields this tidal force. $$ \text { ANS, }-\frac{G M m}{R^{3}}\left(z^{2}-\frac{1}{2} x^{2}-\frac{1}{2} y^{2}\right) $$ In terms of the Legendre polynomials of Chapter 12 this becomes $$ -\frac{G M m}{R^{3}} r^{2} P_{2}(\cos \theta) $$

Using Maxwell's equations, show that for a system (steady current) the magnetic vector potential A satisfies a vector Poisson equation $$ \nabla^{2} \mathbf{A}=-\mu \mathbf{J} $$ provided we require \(\nabla \cdot \mathbf{A}=0\).

Given a vector \(\mathbf{t}=-\hat{\mathrm{x}} y+\hat{\mathrm{y}} x\), with the help of Stokes's theorem, show that the integral around a continuous closed curve in the \(x y\) -plane $$\frac{1}{2} \oint \mathrm{t} \cdot d \lambda=\frac{1}{2} \oint(x d y-y d x)=A,$$ the area enclosed by the curve.
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