Chapter 6, Conductors and Dielectrics Video Solutions, Competitive Physics: Thermodynamics, Electromagnetism and Relativity

Problem 1

A conducting sphere that is currently carrying a total charge $q$ is enclosed by a neutral spherical shell of inner radius $r_1$ and outer radius $r_2$. Both the sphere and the shell are electrically isolated from the rest of the world. A thin wire is then used to connect the sphere and the inner surface of the shell. Determine the final distribution of charge everywhere.

Vishal Gupta

Numerade Educator

01:43

Problem 2

A parallel-plate capacitor is formed by two identical conducting plates, of a large common area, separated by a small distance. One plate has total charge $Q_1$ and the other has total charge $Q_2$. Though the plates are thin, they each still have two surfaces. Neglecting edge effects, calculate the surface charge density on all four surfaces at equilibrium.

Satpal Satpal

Numerade Educator

02:14

Problem 3

A charge $q$ of mass $m$ is placed at a height $h$ above a grounded, infinite conducting plane. If it is then released from rest, determine the time taken for the charge to collide with the plane.

Manik Pulyani

Numerade Educator

01:55

Problem 4

An electrically isolated conducting cube of side length $l$ is neutral. It contains an arbitrary number of cavities that enclose a total amount of charge $Q$. Determine the electric flux cutting across a $l \times l$ plane parallel to and directly above a face of the cube. There are no charges outside the cube.

Aja S

Numerade Educator

03:35

Problem 5

A spherical conducting sphere of radius $R$ is maintained at a potential $V_0$ relative to infinity. It contains an arbitrary number of cavities that enclose arbitrary amounts of charge but not the center of the sphere. If a point charge $Q$ is placed at a distance $d$ from the center of the sphere, outside the sphere, determine the total charge residing on the exterior surface of the sphere at equilibrium.

Sri Datta Vikas Buchemmavari

Numerade Educator

04:07

Problem 6

Considering the problem of a charge $q_1$ that is brought to a distance $r$ from the center of a grounded spherical shell of radius $R$, what is the appropriate image charge configuration if the shell was not grounded and had a net charge $q_3$ instead? What is the force on $q_1$ due to the shell?

Linda Winkler

Numerade Educator

05:15

Problem 7

A semi-infinite conducting plate covers the entire yz-plane in the region $z \geq 0$ while another semi-infinite plate covers the entire $x y$-plane in the region $x \geq 0$. A charge $q$ is placed at $(l, 0, l)$. What is the corresponding image charge configuration and the force on $q$ due to the conducting plates?

Linda Winkler

Numerade Educator

04:19

Problem 8

Determine the charge distribution on an electrically isolated, neutral conducting sphere placed in a region with a uniform external electric field $\boldsymbol{E}$.

Keshav Singh

Numerade Educator

03:02

Problem 9

Define the origin at the center of a neutral conducting sphere of radius $R$ and set up a Cartesian coordinate system. An idealized dipole with dipole moment $\boldsymbol{p}=p \hat{\boldsymbol{k}}$ lies at a positive z-coordinate $r \gg R$. Determine the approximate force experienced by this dipole.

Vishal Gupta

Numerade Educator

04:46

Problem 10

In this problem, we shall directly construct the solution to the charge distribution induced on an infinite conducting cylinder of radius $R$ placed in an electric field $\boldsymbol{E}$ perpendicular to its cylindrical axis. The conducting cylinder is neutral and electrically isolated. Firstly, prove that the electric field is uniform within the overlapping region of two infinite, parallel cylinders of radius $R$ and volume charge densities $\rho$ and $-\rho$. By tweaking this set-up to satisfy the boundary conditions imposed by a particular uniqueness theorem, find the charge distribution on the surface of the conducting cylinder in the original problem.

Suhas Katkar

Numerade Educator

02:28

Problem 11

Prove that if you specify the charge density $\rho(x, y, z)$ in a volume $\Omega$ and the normal derivative of the potential $\frac{\partial V}{\partial n}$ everywhere on the surface $\mathrm{S}$ bounding $\Omega$, the electric field within $\Omega$ is uniquely determined. You do not need to be completely rigorous - an intuitive explanation is fine (e.g. by considering electric field lines).

Eduard Sanchez

Numerade Educator

03:00

Problem 12

Our objective is to determine the capacitances per unit length of the following two systems: (1) two long cylindrical conductors and (2) a long cylindrical conductor and an infinite conducting plane.
(a) In three-dimensional Cartesian coordinates, two infinitely long lines at $x=-a$ and $x=a$ with $y=0$ carry linear charge densities $\lambda$ and $-\lambda$ respectively. Show that the equipotential surfaces of this set-up are infinitely long cylinders. Define the potential at the origin as zero.
(b) Consider a set-up with two infinitely long cylindrical conductors with radius $R$ and their cylindrical axis along lines $x=-b$ and $x=b(b>R)$ with $y=0$. By maintaining the conductors at $x=-b$ and $x=b$ at potentials $V_0$ and $-V_0$ relative to the origin, determine the capacitance per unit length of this system of conductors in light of the previous result.
(c) An infinitely long cylindrical conductor of radius $R$ is now placed with its center a distance $b>R$ above an infinite conducting plane that is grounded. The cylinder is parallel to the plane. Determine the capacitance of this system, per unit length of the cylinder.

Hubert Agamasu

Numerade Educator

01:45

Problem 13

A capacitor with capacitance $C_1$ is initially charged with $q$ amount of charge. Then, its ends are connected via long conducting wires to an initially neutral conductor of capacitance $C_2$. Calculate the energy loss in connecting the capacitors when the system has equilibrated.

Keshav Singh

Numerade Educator

03:38

Problem 14

A cylindrical capacitor has an inner conductor of variable radius $a>0$ and an outer conductor of fixed radius $b$. If the breakdown electric field is given by $E_b$ (a fixed value), determine the relation between $a$ and $b$ such that the capacitor is able to store the most energy. What if we wish to maximize the potential difference between the two conductors?

Suhas Katkar

Numerade Educator

03:07

Problem 15

Two large, electrically isolated capacitor plates of area $A$ and charge densities $\sigma$ and $-\sigma$ are oriented parallel to each other. Suppose that one of the plates is fixed. Determine the work done by an external force in increasing the plate separation by $x$, by pushing away the other plate without increasing its kinetic energy. Do this in three ways: by directly calculating work from force, considering the energy density of the electric field and the potential energy stored in a capacitor.

Manish Kumar ( Iit K )

Numerade Educator

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Problem 16

A conducting sphere of radius $r_1$ is enclosed by a thin concentric conducting shell of radius $r_2>r_1$. Both conductors are grounded. A charge $q$ is now placed between the sphere and the shell at a distance $d\left(r_1<d<r_2\right)$ from the common center. Determine the total charges induced on the sphere and the shell with and without Green's reciprocity theorem.

Gregory Devenport

Numerade Educator

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Problem 17

Consider two concentric spherical shells of radii $r_1$ and $r_2$ with $r_1<r_2$. If $q$ and $-q$ amounts of charge are uniformly spread on the inner and outer shells respectively, determine the energy stored in this capacitor.
Now, consider a separate problem where a point charge $-q$ is located at the center of a spherical, conducting shell of inner and outer radii $r_1$ and $r_2$. The conducting shell is initially neutral and is electrically isolated. Determine the external work done in moving the point charge $-q$ through a narrow hole drilled in the shell to infinity, without a change in kinetic energy. Try to use the previous result.

Lainey Roebuck

Numerade Educator

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Problem 18

Find the resultant capacitance of a "parallel"-plate capacitor if one of the plates were to be tilted slightly at an angle $\theta \ll 1$ with respect to the horizontal. Each place has surface area $A$, horizontal length $l$ and width $w$. The smallest vertical distance between the plates is $d$.

Eduard Sanchez

Numerade Educator

01:37

Problem 19

Determine the equivalent capacitance of a parallel-plate capacitor, with plates of length $l$, width $w$ (directed into the page) and plate separation $h(h \ll l$ and $h \ll w)$, that is half-filled with a triangular dielectric of permittivity $\varepsilon_1$.
(GRAPH CAN'T COPY)

Sheh Lit Chang

University of Washington

09:54

Problem 20

A square parallel-plate capacitor with dimensions $a \times a$ and a separation $d$ is placed inside a beaker of water with density $\rho$ and dielectric constant $\kappa$. Two edges of each plate are aligned with the vertical. A battery is connected to the capacitor such that a constant potential difference $V$ is maintained across its plates. The water between the plates rises up to a height $h$ above the water level in the beaker. Neglecting capillary effects, determine $h$.

Mohammad Mehran

Numerade Educator

04:03

Problem 22

A dielectric with dielectric constant $\kappa$, that is initially neutral everywhere, fills all space. If a spherical cavity of radius $R$ is carved and a uniform external electric field $\boldsymbol{E}_0$ permeates all space, determine the net electric field everywhere.

Khoobchandra Agrawal

Numerade Educator

03:48

Problem 23

An ideal dipole with dipole moment $p$ is embedded at the center of a spherical dielectric with permittivity $\kappa$ and radius $R$. Determine the resultant electric field in all space. In solving this problem, model the ideal dipole as two opposite point charges with a small separation. ${ }^{11}$

Mayukh Banik

Numerade Educator

09:34

Problem 24

An electrically isolated, spherical conductor, of radius $R$ and carrying a total charge $Q$, is centered about the origin. The region $z \geq 0$ (excluding the conductor) is vacuum while the region $z<0$ (excluding the conductor) is filled with a dielectric with permittivity $\varepsilon$. By trying a solution of the form $V(r)=\frac{A}{r}$ (where $A$ is a constant to be determined) for the potential outside the conductor as a function of radial distance from the origin, determine the potential in all space. Assume that the conditions of this problem are set up such that the potential is unique.

Khoobchandra Agrawal

Numerade Educator

Problem 25

A dielectric medium of dielectric constant $\kappa$ fills the entire region $z \leq 0$. A point charge $q$ is placed at a positive z-coordinate $z=d$ (the region $z>0$ is in vacuum). Determine the force experienced by $q$.

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02:19

Problem 212

Two large parallel plates with a narrow separation carry fixed, uniform surface charge densities $\sigma$ and $-\sigma$ respectively. Two large slabs with identical surface areas as the plates and permittivities $\varepsilon_1$ and $\varepsilon_2$ are then slotted between the plates such that the gap within the plates is filled completely. If the slab with permittivity $\varepsilon_1$ is closer to the plate with surface charge density $\sigma$, determine the electric displacements, electric fields and polarizations in the two slabs. Finally, determine the bound charges everywhere.

Manik Pulyani

Numerade Educator