The Dipole Far Field and the Field of a Charged Arc

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1. Question 1

   A charge $+q$ sits at $(0, A)$ and $-q$ at $(0, -A)$, with $A = 0.040\ \text{nm}$. A point P is $r = 2.0\ \text{nm}$ from the origin at $\theta = 60\degree$ from the $+y$ axis, and $r_-$ is its distance from $-q$. Using the first-order binomial approximation, find $(r/r_-)^3$ to three decimal places.

   $(r/r_-)^3 \approx$

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2. Question 2

   Use $k_e = 8.99\times10^{9}\ \text{N}\,\text{m}^2/\text{C}^2$. A dipole of moment $p = 2.0\times10^{-29}\ \text{C}\,\text{m}$ is at the origin. At $r = 4.0\ \text{nm}$, $\theta = 60\degree$ from $\vec{\mathbf{p}}$, find $E_r$ and $E_\theta$ from the far-field formulas, in units of $10^{6}\ \text{N/C}$ to three significant figures.

   $E_r =$×10⁶ N/C

   $E_\theta =$×10⁶ N/C

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3. Question 3

   Use $k_e = 8.99\times10^{9}\ \text{N}\,\text{m}^2/\text{C}^2$. A dipole of moment $p = 6.0\times10^{-30}\ \text{C}\,\text{m}$ is at the origin. Find the far-field magnitude at $r = 3.0\ \text{nm}$, $\theta = 30\degree$ from $\vec{\mathbf{p}}$, in units of $10^{6}\ \text{N/C}$ to three significant figures.

   $|\vec{\mathbf{E}}| =$×10⁶ N/C

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4. Question 4

   At a point $\theta = 50\degree$ from a dipole's axis, far from the dipole, find the angle between $\vec{\mathbf{E}}$ and $\hat{\mathbf{r}}$, in degrees to one decimal place.

   °

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5. Question 5

   Use $k_e = 8.99\times10^{9}\ \text{N}\,\text{m}^2/\text{C}^2$. An arc of radius $R = 0.15\ \text{m}$ with $\lambda = 4.0\ \text{nC/m}$ runs from $-30\degree$ to $+30\degree$ about the $x$-axis. Find the field magnitude at its centre, in N/C to three significant figures.

   $E =$N/C

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