Question

Part A: What is the x-component of vector E⃗ of the figure in terms of the angle θ and the magnitude E ? (Express your answer in terms of the variables θ and E )

Part B: What is the y-component of vector E⃗ of the figure in terms of the angle θ and the magnitude E ? (Express your answer in terms of the variables θ and E )

Part C: For the same vector, what is the x-component in terms of the angle ϕ and the magnitude E ? (Express your answer in terms of the variables ϕ and E )

Part D: For the same vector, what is the y-component in terms of the angle ϕ and the magnitude E ? (Express your answer in terms of the variables ϕ and E )

Answer 1

Concepts and reason

The concept required to solve the given problem is field components along horizontal and vertical axis.

Initially, consider $\theta$ as a base angle and find the components of $\vec E$ along x and y axis. In the next step, consider $\phi$ as a base angle and find the components of $\vec E$ along x and y axis.

Fundamentals

Consider a vector $\vec A$ that makes some angle $\theta$ with the x axis. The following figures illustrates that:

The x component of vector $\vec A$ is,

${A_{\rm{x}}} = A\cos \theta$

The y component of vector $\vec A$ is,

${A_{\rm{y}}} = A\sin \theta$

(A)

The following figure shows the x and y components of $\vec E$ with $\theta$ as a base angle.

Here, ${E_{\rm{x}}}$ is the x component of $\vec E$ and ${E_{\rm{y}}}$ is the y component of $\vec E$ .

The x component of $\vec E$ is,

$\begin{array}{c}\\{E_{\rm{x}}} = E\left( { - \cos \theta } \right)\\\\ = - E\cos \theta \\\end{array}$

Here, E is the magnitude of $\vec E$ .

The negative sign represents that the component lies in the negative x direction.

(B)

The following figure shows the x and y components of $\vec E$ with $\theta$ as a base angle.

Here, ${E_{\rm{x}}}$ is the x component of $\vec E$ and ${E_{\rm{y}}}$ is the y component of $\vec E$ .

The y component of $\vec E$ is,

$\begin{array}{c}\\{E_{\rm{x}}} = E\left( {\sin \theta } \right)\\\\ = E\sin \theta \\\end{array}$

Here, E is the magnitude of $\vec E$ .

(C)

The relation between angle $\phi$ and $\theta$ is,

$\begin{array}{c}\\\theta + \phi = 90^\circ \\\\\theta = 90^\circ - \phi \\\end{array}$

The x component of $\vec E$ is,

$\begin{array}{c}\\{E_{\rm{x}}} = - E\cos \theta \\\\ = - E\cos \left( {90^\circ - \phi } \right)\\\\ = - E\sin \phi \\\end{array}$

Here, E is the magnitude of $\vec E$ .

The negative sign represents that the component lies in the negative x direction.

(D)

The relation between angle $\phi$ and $\theta$ is,

$\begin{array}{c}\\\theta + \phi = 90^\circ \\\\\theta = 90^\circ - \phi \\\end{array}$

The y component of $\vec E$ is,

$\begin{array}{c}\\{E_{\rm{y}}} = E\sin \theta \\\\ = E\sin \left( {90^\circ - \phi } \right)\\\\ = E\cos \phi \\\end{array}$

Here, E is the magnitude of $\vec E$ .

Ans: Part A

The x component of $\vec E$ is $- E\cos \theta$ .

[Answer Choice]

Part B

The y component of $\vec E$ is $E\sin \theta$ .

[Answer Choice]

Part C

The x component of $\vec E$ in terms of $\phi$ is $- E\sin \phi$ .

[Answer Choice]

Part D

The y component of $\vec E$ in terms of $\phi$ is $E\cos \phi$ .

[Answer Choice]