Chapter 5, Electrostatics Video Solutions, Competitive Physics: Thermodynamics, Electromagnetism and Relativity

Problem 1

Two point charges, which are not necessarily of the same charge, are placed along the positive $\mathrm{x}$-axis. The electric potential is found to tend to positive infinity when approaching $x=0$. Next, it is known that the electric potential is zero at two points on the positive $\mathrm{x}$-axis, where $x_0$ is the larger $\mathrm{x}$-coordinate of the two. Lastly, the electric potential at $x=\alpha x_0$ is a local minimum along the $\mathrm{x}$-direction where $\alpha$ is a positive constant. What can you say about the signs and relative magnitude of the charges? Determine the distance $d$ between the charges.

Sophie S

Numerade Educator

03:10

Problem 2

Three identical charges of mass $m$ and charge $q$ are initially positioned such that they form the vertices of an equilateral triangle with sides $l_0$. If they were initially stationary, determine the velocities of the charges when their relative separation becomes $l$ afterwards.

Katie Mcalpine

Numerade Educator

04:00

Problem 3

A negative point charge $-q$ of mass $m$ is currently orbiting a fixed, positive charge $Q$ at a radius of rotation $r_0$. Now, the orbiting charge suddenly disintegrates such that it ejects half of its mass in the radial direction at a negligible velocity, relative to the rest of the charge. However, the leftover mass still retains the entire charge $-q$. Determine the minimum distance between the leftover charge and $Q, r$, in the motion thereafter while neglecting any gravitational effects.

Christopher Provencher

Numerade Educator

03:28

Problem 4

A cube of length $l$ possesses a uniform volume charge density $\rho$. Find the ratio of the electric potential at one of its vertices to that at its center.

Khoobchandra Agrawal

Numerade Educator

01:37

Problem 5

A point charge $q$ is placed at one of the vertices of an imaginary cubic Gaussian surface. Find the electric flux through one of the faces of the cube that is non-zero.

Suhas Katkar

Numerade Educator

02:59

Problem 6

A charge $q$ is placed off-center in an imaginary sphere. Determine the total electric flux cutting the spherical cap (in bold) depicted in the figure below, that is characterized by the distances $a$ and $h$, for $h=0$. Next, solve for the electric flux across the spherical cap for general $a$ and $h$ and check that your result yields the right answer for the previous limiting case.
(GRAPH CAN'T COPY)

Ajay Singhal

Numerade Educator

03:02

Problem 7

The six faces of an insulating cube of edge length $l$ are coated with a uniform surface charge density $\sigma$. Determine the force experienced by one face of the cube.

Dr. Rajveer Singh

Numerade Educator

07:44

Problem 8

Determine the electric field at a point $\mathrm{P}$ located at a height $\frac{l}{2 \sqrt{6}}$ above the centroid of an equilateral triangle with edge length $l$ and uniform surface charge density $\sigma$.

Deepak Kohli

Numerade Educator

01:06

Problem 9

Every black tile of an insulating chessboard of dimensions $l \times l$ is painted with a uniform surface charge density $\sigma$ while every white tile is entirely neutral. Determine the electric field at a point $\frac{l}{2}$ above the center of the chessboard without any integration.
(GRAPH CAN'T COPY)

Raj Bala

Numerade Educator

09:48

Problem 10

Prove that the direction of the electric field at an arbitrary point $\mathrm{P}$ due to a finite line charge, with uniform linear charge density $\lambda$ and ends at $A$ and $B$, always bisects $\angle A P B$. Furthermore, show that the magnitude of the electric field at $\mathrm{P}$ is
$$
E=\frac{\lambda}{2 \pi \varepsilon_0 h} \sin \alpha,
$$
where $h$ is the perpendicular distance between $\mathrm{P}$ and the line and $\alpha=$ $\frac{1}{2} \angle A P B$. Even though we have previously derived the electric field of a line charge, do not overlook this part but rather, search for a different method that directly proves the above properties. Next, determine the electric field vector at $(a, a)$ in the xy-plane due to two line charges, with uniform linear charge density $\lambda$, that lie from $(a, 0)$ to $(+\infty, 0)$ and from $(0, a)$ to $(0,+\infty)$.

Deven Gill

Numerade Educator

View

Problem 11

A hydrogen atom is made up of a proton and an electron. The proton may be regarded as a point charge $q$ at $r=0$, the center of the atom. Meanwhile, the motion of the electron causes its charge to be "smeared out" into a spherically symmetric distribution around the proton, such that the electron is equivalent to a charge density
$$
\rho(r)=-\frac{q}{\pi a_0^3} e^{-\frac{2 r}{a_0}},
$$
where $a_0$ is a constant known as the Bohr radius and $r$ is the radial distance from the center. Note that $e$ is Euler's number and not the charge of the electron.
(a) Find the total amount of the hydrogen atom's charge that is enclosed within a sphere of radius $r$, centered about the proton. Check your answer for the limit $r \rightarrow \infty$ and explain why it makes sense.
(b) Find the expression for the electric field strength $E$ as a function of $r$.
(c) Find the expression for the electric potential $V$ as a function of $r$.

Victor Salazar

Numerade Educator

02:10

Problem 12

Two charges $q>0$ and $-q$ are located along the $\mathrm{x}$-axis at $(-d, 0,0)$ and $(d, 0,0)$. Determine the radial distance from the origin that a field line emanating from $q$ at an angle $\alpha$ with respect to the positive $\mathrm{x}$-axis intersects with the yz-plane.

Khoobchandra Agrawal

Numerade Educator

01:37

Problem 13

(a) Consider the isolated system of a single point charge $q$. Determine the average electric field vector over the surface of an imaginary sphere, centered about an arbitrary point and possessing an arbitrary radius $r$, that encloses the charge $q$.
(b) Using the same system as above, determine the average electric field vector over the surface of an imaginary sphere, centered about an arbitrary point and possessing an arbitrary radius $r$, that does not enclose the charge $q$. Let the vector pointing from $q$ to the center of the sphere be $\boldsymbol{R}$.
(c) Show that for an arbitrary system of point charges, the average potential over the surface of an imaginary sphere, centered about an arbitrary point and possessing an arbitrary radius $r$, that does not enclose any charge is identical to the potential at the center of the sphere; $V_{a v g}=$ $V_{\text {center. }}$. Explain why this proves Earnshaw's Theorem.
(d) More generally, show that for an arbitrary system of point charges, the average potential over the surface of an imaginary sphere, centered about an arbitrary point and possessing an arbitrary radius $r$, is $V_{a v g}=V_{\text {center }}+\frac{q_{\text {enc }}}{4 \pi \varepsilon_0 r}$ where $V_{\text {center }}$ is the potential at the center of the sphere and $q_{\text

Dominador Tan

Numerade Educator

05:21

Problem 14

Determine the electric field at a point of height $h$ above the center of a thin, circular disk that has a uniform surface charge density $\sigma$ and radius $R$. The point of concern is along the axis of the disk.
In light of the above result, determine the electric field, due to the truncated cone with a uniform volume charge density $\rho$ shown in the figure on the next page, at the vertex of the original cone.

Sri Datta Vikas Buchemmavari

Numerade Educator

05:57

Problem 15

A hole of radius $R$ is carved out of a thin infinite plane with a positive surface charge density $\sigma$ that is uniform. Place a charge $q$ at the center of the hole. Neglecting all gravitational effects, show that the center of the hole corresponds to an equilibrium position. Determine the stability of the equilibrium of charge $q$ with mass $m$ when it is slightly displaced in the direction normal to the plane (you have to consider different values of $q$ ). If the equilibrium is stable, determine the angular frequency of small oscillations that $q$ exhibits. You may find the result of the previous problem to be useful.

Jonathan Everett

Numerade Educator

07:14

Problem 16

The cylindrical axis of a cylinder with length $l$ and radius $R$ is aligned with the z-axis. The centers of the bases of the cylinder are at $z=0$ and $z=l$. If the cylinder has a uniform charge density $\rho$, determine the electric field everywhere along the cylindrical axis, both inside and outside of the cylinder.

Keshav Singh

Numerade Educator

06:24

Problem 17

A solid sphere of radius $R$ is made up of an insulating material and has a volume charge density $\rho$. A spherical cavity of radius $a$ is removed from the sphere. The center of the cavity is at a position $\boldsymbol{d}$ with respect to the center of the sphere. Determine the electric field everywhere within the cavity.

Nathan Silvano

Numerade Educator

06:38

Problem 18

Two identical thin rods of length $l$ have equal uniform linear charge density $\lambda$. They both lie along the $\mathrm{x}$-axis with their centers separated by a distance $d>l$. Determine the magnitude of the Coulomb force exerted on the right rod by the left rod at this instance. Check if your answer returns the correct limit when $d \gg l$.

Dading Chen

Numerade Educator

05:59

Problem 19

A square conducting plate of side length $2 a$, centered about the origin in the xy-plane, is charged with a uniform surface charge density $\sigma$.
(a) Prove that following integral where $z$ is independent of $x$ and $y$ :
$$
\int_0^a \int_0^a \frac{d x d y}{\left(x^2+y^2+z^2\right)^{\frac{3}{2}}}=\frac{1}{z} \tan ^{-1} \frac{a^2}{z \sqrt{2 a^2+z^2}} .
$$
(b) Determine the electric field $\boldsymbol{E}$ at $(0,0, z)$ due to this surface charge distribution. Find the limits where $z \rightarrow 0^{+}$and $z \rightarrow 0^{-}$and explain why they make sense.
Now, in addition to the previous charged square plate, there is another square plate of the same size, parallel to the xy-plane and centered at $(0,0, d)$. This additional plate is uniformly charged with a surface charge density $-\sigma$.
(c) Determine the electric field at $(0,0, z)$ for all $z$ due to the uniform charge distributions on both plates (we assume, albeit incorrectly, that they stay uniform in the presence of each other).
(d) Assuming $d<<a$, find the asymptotic solution to the previous field for all $z$. As an aside, the charge distributions on both plates indeed remain uniform in this limit as the plates are effectively infinitely large as compared to the separation between them.
(e) Based on your previous answer, determine the potential difference $V$ between the two plates. The capacitance of the two plates is defined as
$$
C=\left|\frac{Q}{V}\right|
$$
where $Q$ is the total charge on either plate. Find $C$ and state the variables it depends on.

Khoobchandra Agrawal

Numerade Educator

03:22

Problem 20

An insulating sphere of radius $R$ is coated with a uniform surface charge density $\sigma$ on its exterior surface. Suppose that we cut off a spherical cap corresponding to a half-angle $\alpha$ from a certain axis (i.e. the cap has base radius $R \sin \alpha$ ) and remove the rest of the sphere. Determine the electric field due to the cap at the center of the original sphere. Thus, state the electric field at the center of an insulating hemisphere with surface charge density $\sigma$ coated over its curved surface. Finally, using this result for a hemisphere, determine the electric field at the center of the original sphere if a spherical wedge of half-angle $\alpha$ (a slice of watermelon) is extracted from the sphere and the rest of the sphere is removed instead.

Suzanne W.

Numerade Educator

05:17

Problem 21

Three line charges, each of length $L$, are arranged in the form of an equilateral triangle. The line charges carry uniform charge densities $2 \lambda, \lambda$ and $\lambda$.
(a) Determine the electric field at the centroid of the triangle.
(b) Determine the electric potential at the centroid of the triangle.
(c) Find a point inside the triangle where the electric field is zero.

Ben Nicholson

Numerade Educator

06:16

Problem 22

Determine the electric potential at the rim of an insulating disk of radius $R$ and uniform surface charge density $\sigma$. Hint: adopt polar coordinates about a point on the rim for the integration. In light of your result, derive the electric potential energy stored in the disk.

Mohit Khurana

Texas A&M University

01:40

Problem 23

Firstly, determine the potential at the vertex which is sandwiched between the two equal edges (which subtend an angle $2 \alpha$ ) of an isosceles triangle with uniform surface charge density $\sigma$, if the height of the triangle from this vertex is $h$. Using this result, determine the potential at the center and thus the potential of a corner of a square with edge length $l$ and a uniform surface charge density $\sigma$.

Manik Pulyani

Numerade Educator

05:06

Problem 24

Consider a charged sphere of radius $R$ whose northern and southern hemispheres carry volume charge densities $\rho_1$ and $\rho_2$ respectively. Determine the force between these hemispheres.

Eduard Sanchez

Numerade Educator

04:07

Problem 25

Determine the force between two uniform and concentric hemispherical shells of radii $R, r<R$ and charges $Q, q$ respectively. The common center and the apexes of the shells are collinear.

Linda Winkler

Numerade Educator