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An electric field in free space is given as \(\mathbf{E}=x \hat{\mathbf{a}}_{x}+4 z \hat{\mathbf{a}}_{y}+4 y \hat{\mathbf{a}}_{z}\). Given \(V(1,1,1)=10 \mathrm{~V}\), determine \(V(3,3,3)\).

Short Answer

Expert verified
Answer: The potential difference at point (3,3,3) is 82 V.

Step by step solution

01

Find \(\mathbf{E} \cdot d \boldsymbol{\ell}\)

We are given the electric field as \(\mathbf{E} = x\hat{\mathbf{a}}_{x} +4 z \hat{\mathbf{a}}_{y} + 4 y \hat{\mathbf{a}}_{z}\). Now we need to find \(d\boldsymbol{\ell}\) which is a differential length vector that lies along the path of integration. A simple choice is to take \(d\boldsymbol{\ell} = dx\hat{\mathbf{a}}_{x}+ dy\hat{\mathbf{a}}_{y}+dz\hat{\mathbf{a}}_{z}\). Thus, the dot product \(\mathbf{E} \cdot d\boldsymbol{\ell}\) can be found as $$ \mathbf{E} \cdot d\boldsymbol{\ell} = x\hat{\mathbf{a}}_{x} \cdot (dx\hat{\mathbf{a}}_{x}+ dy\hat{\mathbf{a}}_{y}+dz\hat{\mathbf{a}}_{z}) +4z\hat{\mathbf{a}}_{y}\cdot(dx\hat{\mathbf{a}}_{x}+ dy\hat{\mathbf{a}}_{y}+dz\hat{\mathbf{a}}_{z}) + 4y\hat{\mathbf{a}}_{z}\cdot (dx\hat{\mathbf{a}}_{x}+ dy\hat{\mathbf{a}}_{y}+dz\hat{\mathbf{a}}_{z}). $$ Now, we know that \(\hat{\mathbf{a}}_i \cdot \hat{\mathbf{a}}_j = \delta_{ij}\), so we get $$ \mathbf{E} \cdot d\boldsymbol{\ell} = x dx+4z dy+ 4y dz. $$
02

Perform the integration

We need to find the potential difference due to electric field along the path from point (1,1,1) to point (3,3,3), so we need to integrate \(\mathbf{E} \cdot d\boldsymbol{\ell}\) with respect to \(x\), \(y\), and \(z\). $$ V(3,3,3)- V(1,1,1)= -\int_{1}^{3} x dx - \int_{1}^{3} 4z dy - \int_{1}^{3} 4y dz. $$ Computing the integrals, we get $$ V(3,3,3)- V(1,1,1)= -\frac{1}{2}(3^{2} - 1^{2}) - 4 \cdot \frac{1}{2}(3^{2} - 1^{2}) - 4 \cdot \frac{1}{2}(3^{2} - 1^{2})=-8-64=-72. $$
03

Calculate \(V(3,3,3)\)

Now, we can find \(V(3,3,3)\) using \(V(1,1,1) = 10\) V. $$ V(3,3,3)= V(1,1,1) - (V(3,3,3)-V(1,1,1))= 10 -(-72)=82 \mathrm{~V}. $$ So, the potential difference at point (3,3,3) is \(V(3,3,3) = 82\)V.

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Key Concepts

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

Electric Field in Free Space
When studying the electric field in free space, we explore the behavior of electric forces in a vacuum or a region devoid of matter. An electric field represents the force experienced by a test charge placed within it and is a vector field characterized by both magnitude and direction.

For example, consider an electric field \textbf{E} given as \textbf{E} = x\(\bf{\hat{a}}_{x}\) + 4z\(\bf{\hat{a}}_{y}\) + 4y\(\bf{\hat{a}}_{z}\). Here, the field components relate to different coordinates, suggesting that the field's strength and direction vary depending on the position in space. To find the electric potential at a point within such a field, we apply the concept of electric potential difference which is the work done per unit charge in moving a test charge within this field.
Vector Calculus in Electromagnetism
Vector calculus is an indispensable tool in electromagnetism because fields and potentials are best described using vectors and vector operations. The dot product, for instance, is a way to quantify the work done by the electric field along a path element \(d\boldsymbol{\ell}\).

In electromagnetism, understanding how to manipulate vector fields through operations such as gradients, divergences, and curls is critical for explaining physical phenomena. For the electric field \(\mathbf{E} = x\hat{\mathbf{a}}_{x} +4 z \hat{\mathbf{a}}_{y} + 4y\hat{\mathbf{a}}_{z}\), a grasp of how these vector operations manifest in the physics is key to understanding the flow of electric charges and the potential effects they produce.
Integration of Vector Fields
Integration is a fundamental method for calculating quantities over specific regions in space or along paths. When integrating a vector field such as the electric field, we often have to deal with line integrals. In the given exercise, the line integral is used to calculate the electric potential difference between two points by integrating the component of the electric field along a chosen path.

The integration \(\int \mathbf{E} \. d\boldsymbol{\ell}\) simplifies greatly due to the orthogonality of unit vectors \(\hat{\mathbf{a}}_{i}\) (they have a dot product of zero unless they are the same), leading to separate integrals for each variable. With parameters such as x, y, and z each varying between two points, calculating the potential difference requires integrating their respective contributions, clearly shown in the steps of the provided solution.

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Most popular questions from this chapter

A copper sphere of radius \(4 \mathrm{~cm}\) carries a uniformly distributed total charge of \(5 \mu \mathrm{C}\) in free space. \((a)\) Use Gauss's law to find \(\mathbf{D}\) external to the sphere. (b) Calculate the total energy stored in the electrostatic field. (c) Use \(W_{E}=\) \(Q^{2} /(2 C)\) to calculate the capacitance of the isolated sphere.

Within the cylinder \(\rho=2,0

Two uniform line charges, \(8 \mathrm{nC} / \mathrm{m}\) each, are located at \(x=1, z=2\), and at \(x=-1, y=2\) in free space. If the potential at the origin is \(100 \mathrm{~V}\), find \(V\) at \(P(4,1,3)\)

Let us assume that we have a very thin, square, imperfectly conducting plate \(2 \mathrm{~m}\) on a side, located in the plane \(z=0\) with one corner at the origin such that it lies entirely within the first quadrant. The potential at any point in the plate is given as \(V=-e^{-x} \sin y .(a)\) An electron enters the plate at \(x=0, y=\pi / 3\) with zero initial velocity; in what direction is its initial movement? ( \(b\) ) Because of collisions with the particles in the plate, the electron achieves a relatively low velocity and little acceleration (the work that the field does on it is converted largely into heat). The electron therefore moves approximately along a streamline. Where does it leave the plate and in what direction is it moving at the time?

A line charge of infinite length lies along the \(z\) axis and carries a uniform linear charge density of \(\rho_{\ell} \mathrm{C} / \mathrm{m}\). A perfectly conducting cylindrical shell, whose axis is the \(z\) axis, surrounds the line charge. The cylinder (of radius \(b\) ), is at ground potential. Under these conditions, the potential function inside the cylinder \((\rhob .(d)\) Find the stored energy in the electric field per unit length in the \(z\) direction within the volume defined by \(\rho>a\), where \(a

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