Chapter 21 — Electric Charge & the Electric Field

12 points · 3 extra credit

Chapter 21 — Electric Charge & the Electric Field

Core: 12 points · Extra credit: 3 points. Show your reasoning; partial credit is given.

Use $k=8.99\times10^{9}\,\text{N}\cdot\text{m}^2/\text{C}^2$.

1. 1. (2 pts ••) Two small spheres carry $q_1=+4.0\,\mu\text{C}$ and $q_2=-7.0\,\mu\text{C}$ and are $30\,\text{cm}$ apart.

   (a) Find the magnitude of the force each exerts on the other.

   (b) A third charge $q_3=+2.0\,\mu\text{C}$ is placed at the midpoint between them. Find the magnitude and direction of the net force on $q_3$.

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2. 2. (3 pts •••) Four equal charges $+q$ sit at the corners of a square of side $L$.

   (a) Find the magnitude and direction of the net force on one corner charge.

   (b) Find the electric field at the center of the square.

   (c) A charge $-Q$ is placed at the center. Is it in equilibrium? Is that equilibrium stable? Justify your answer.

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3. 3. (2 pts ••) A charge $q_1=+5.0\,\text{nC}$ is at the origin and $q_2=-3.0\,\text{nC}$ is at $(4.0\,\text{cm},\,0)$.

   (a) Find the electric field at the point $P=(0,\,3.0\,\text{cm})$ (magnitude and direction).

   (b) Find the force on a $-1.0\,\text{nC}$ charge placed at $P$.

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4. 4. (2 pts ••) An electric dipole has charges $\pm q$ with $q=3.0\,\text{nC}$ separated by $d=2.0\,\text{mm}$, in a uniform field $E=4.0\times10^{5}\,\text{N/C}$.

   (a) Find the maximum torque on the dipole.

   (b) Find the work needed to rotate it from aligned ($\theta=0$) to $\theta=90^\circ$.

   (c) …and from $\theta=0$ to $\theta=180^\circ$.

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5. 5. (2 pts ••) In a simplified inkjet printer, a droplet of mass $m=1.3\times10^{-10}\,\text{kg}$ and charge $q=-1.5\times10^{-13}\,\text{C}$ enters a $1.6\,\text{cm}$-long region of uniform field $E=8.0\times10^{4}\,\text{N/C}$ (pointing down) at a horizontal speed of $20\,\text{m/s}$. Ignoring gravity, find the vertical displacement of the droplet while it travels through the field region.

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6. 6. (1 pt •) A charged particle feels an upward force of $4.0\times10^{-6}\,\text{N}$ in a uniform field of $2.0\times10^{3}\,\text{N/C}$ that points downward. Find the sign and magnitude of its charge.

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7. Extra Credit (3 pts •••)

   A thin nonconducting rod of length $L$ carries uniform linear charge density $\lambda$.

   (a) Starting from Coulomb's law, integrate to find the electric field at a point on the rod's axis a distance $a$ beyond the near end.

   (b) Do the same for a point a perpendicular distance $y$ from the rod's midpoint, and show your result reduces to the infinite-line field $E=\dfrac{\lambda}{2\pi\varepsilon_0 y}$ as $L\to\infty$.

   (c) Show that in the appropriate far-field limit each result reduces to that of a point charge.

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