Chapter 23 — Electric Potential

12 points · 3 extra credit

Chapter 23 — Electric Potential

Core: 12 points · Extra credit: 3 points. Show your reasoning; partial credit is given. Use $k=8.99\times10^{9}\,\text{N}\cdot\text{m}^2/\text{C}^2$, $e=1.60\times10^{-19}\,\text{C}$, $m_e=9.11\times10^{-31}\,\text{kg}$, $m_p=1.67\times10^{-27}\,\text{kg}$.

1. 1. (2 pts ••) A charge $q_1=+4.0\,\text{nC}$ is at the origin and $q_2=-3.0\,\text{nC}$ is at $x=0.50\,\text{m}$.

   (a) Find the electric potential at the midpoint between them ($x=0.25\,\text{m}$), taking $V=0$ at infinity.

   (b) Find the work required to bring a $+2.0\,\text{nC}$ charge from infinity to that midpoint.

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2. 2. (2 pts ••) A proton is released from rest at a distance of $2.0\,\text{cm}$ from a fixed point charge $Q=+5.0\,\text{nC}$. Find its speed when it has moved to $10.0\,\text{cm}$ from the charge.

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3. 3. (2 pts ••) Two large parallel plates are $5.0\,\text{mm}$ apart with a potential difference of $200\,\text{V}$ between them.

   (a) Find the magnitude of the electric field between the plates.

   (b) An electron starts from rest at the negative plate. Find its speed when it reaches the positive plate.

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4. 4. (2 pts ••) A conducting sphere of radius $5.0\,\text{cm}$ carries a charge of $+8.0\,\text{nC}$. Take $V=0$ at infinity.

   (a) Find the potential at the surface.

   (b) Find the potential at $r=15\,\text{cm}$.

   (c) Find the potential at the center of the sphere.

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5. 5. (3 pts •••) In a region of space the electric potential is $V(x,y,z)=3x^2y-2z$, with $V$ in volts and $x,y,z$ in meters.

   (a) Find the electric field $\mathbf{E}(x,y,z)$ from the potential gradient.

   (b) Evaluate $\mathbf{E}$ at the point $(2,\,-1,\,3)\,\text{m}$.

   (c) Find the magnitude of $\mathbf{E}$ there.

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6. 6. (1 pt •) An alpha particle (charge $+2e$) is accelerated from rest through a potential difference of $1.0\,\text{kV}$. Find its kinetic energy in joules.

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7. Extra Credit (3 pts •••)

   A thin disk of radius $R$ carries a uniform surface charge density $\sigma$.

   (a) By integrating the contributions of thin ring elements, find the electric potential $V(x)$ on the disk's axis a distance $x$ from its center (take $V=0$ at infinity).

   (b) From $V(x)$, find the axial field $E_x(x)$.

   (c) Show that for $x\gg R$ your result reduces to that of a point charge, and state what $E_x$ approaches for $x\ll R$.

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