Electric Fields

In Millikan’s experiment, an oil drop of radius $\mathbf{1}\mathbf{.64}\mathbf{ }\mathbf{\text{μm}}$ and density  $\mathbf{0}\mathbf{.851}\mathbf{ }{\mathbf{\text{g/cm}}}^{\mathbf{\text{3}}}$is suspended in chamber C (Fig. 22-16) when a downward electric field of $\mathbf{1}\mathbf{.92}\mathbf{ }\mathbf{\times }\mathbf{ }{\mathbf{10}}^{\mathbf{5}}\mathbf{\text{N/C}}$ is applied. Find the charge on the drop, in terms of e.

An electron with a speed of $\mathbf{5}\mathbf{.00}\mathbf{ }\mathbf{\times }\mathbf{ }{\mathbf{10}}^{\mathbf{8}}\mathbf{\text{cm/s}}$  enters an electric field of magnitude $\mathbf{1}\mathbf{.00}\mathbf{ }\mathbf{\times }{\mathbf{10}}^{\mathbf{3}}\mathbf{\text{N/C}}$ , traveling along a field line in the direction that retards its motion. (a) How far will the electron travel in the field before stopping momentarily, and (b) how much time will have elapsed? (c) If the region containing the electric field is $\mathbf{8}\mathbf{.00}\mathbf{\text{\hspace{0.17em}mm}}$  long (too short for the electron to stop within it), what fraction of the electron’s initial kinetic energy will be lost in that region?

A charged cloud system produces an electric field in the air near Earth’s surface. A particle of charge $\mathbf{-}\mathbf{2}\mathbf{.0}\mathbf{ }\mathbf{\times }\mathbf{ }{\mathbf{10}}^{\mathbf{-}\mathbf{9}}\mathbf{C}$ is acted on by a downward electrostatic force of  $\mathbf{3.0}\mathbf{ }\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathbf{N}$ when placed in this field. (a) What is the magnitude of the electric field? What are the (b) magnitude and (c) direction of the electrostatic Force ${\overrightarrow{\mathbf{f}}}_{\mathbf{el}}$ on the proton placed in this field? (d)What is the magnitude of the gravitational force ${\overrightarrow{\mathbf{f}}}_{\mathbf{g}}$ on the proton? (e) What is the ratio ${\mathbf{F}}_{\mathbf{el}}\mathbf{ }\mathbf{/}{\mathbf{F}}_{\mathbf{g}}$ in this case?

An electron is released from rest in a uniform electric field of magnitude$\mathbf{2}\mathbf{.00}\mathbf{ }\mathbf{\times }\mathbf{ }{\mathbf{10}}^{\mathbf{4}}\mathbf{\text{\hspace{0.17em}N/C}}$. Calculate the acceleration of the electron. (Ignore gravitation.)

Question: An alpha particle (the nucleus of a helium atom) has a mass of $\mathbf{6}\mathbf{.}\mathbf{64}\mathbf{\times }\mathbf{ }{\mathbf{10}}^{\mathbf{-}\mathbf{27}}\mathbf{kg}$ and a charge of +2e. What are the (a) magnitude and (b) direction of the electric field that will balance the gravitational force on the particle?

An electron on the axis of an electric dipole is $\mathbf{25}\mathbf{\text{ nm}}$ from the center of the dipole.What is the magnitude of the electrostatic force on the electron if the dipole moment is$\mathbf{3}\mathbf{.6}\mathbf{ }\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{29}}\mathbf{\text{C}}\mathbf{\cdot }\mathbf{\text{m}}$? Assume that  $\mathbf{25}\mathbf{\text{ nm}}$is much larger than the separation of the charged particles that form the dipole.

Question: An electron is accelerated eastward at $1.80\times {10}^9 m/s^2$ by an electric field. Determine the field (a) magnitude and (b) direction.

Question: Beams of high-speed protons can be produced in “guns” using electric fields to accelerate the protons. (a) What acceleration would a proton experience if the gun’s electric field were $2.00\times {10}^4 N/C$ ? (b) What speed would the proton attain if the field accelerated the proton through a distance of 1.00 cm?

Question: In Fig. 22-59, an electron (e) is to be released from rest on the central axisof a uniformly charged disk of radius R. The surface charge density on the disk is$+4.00 mC/m^2$. What is the magnitude of the electron’s initial acceleration if it is released at a distance (a) R, (b) R/100 , and (c) R /1000 from the center of the disk? (d) Why does the acceleration magnitude increase only slightly as the release point is moved closer to the disk?

Question: A 10.0 g block with a charge of$+8.00 \times {10}^{-5}C$  is placed in an electric field $\overrightarrow{E}=\left(3000\hat{i}-600\hat{j}\right)N/C$. What are the (a) magnitude and (b) direction (relative to the positive direction of the x axis) of the electrostatic force on the block? If the block is released from rest at the origin at time t = 0, what is its (c) x and (d) y coordinates at t =3.00s?

Question: At some instant the velocity components of an electron moving between two charged parallel plates are $V_X=1.5\times {10}^{5}m/s$  and$V_y=3.0\times {10}^{3}m/s$. Suppose the electric field between the plates is uniform and given by $\overrightarrow{F}=\left(120N/C\right)\hat{j}$ .In unit-vector notation, what are (a) the electron’s acceleration in that field and (b) the electron’s velocity when its x coordinate has changed by 2.0 cm?

Question: An electron enters a region of uniform electric field with an initial velocity of 40km/s in the same direction as the electric field, which has magnitude E =50N/C. (a) What is the speed of the electron 1.5ns after entering this region? (b) How far does the electron travel during 1.5ns the interval?

Two large parallel copper plates are $\mathbf{5}\mathbf{.0}\mathbf{ }\mathbf{\text{cm}}$ apart and have a uniform electric field between them as depicted in Fig. 22-60. An electronis released from the negative plate at the same time that a proton is released from the positive plate. Neglect the force of the particles on each other and find their distance from the positive plate when they pass each other. (Does it surprise you that you need not know the electric field to solve this problem?)

In Fig. 22-61, an electron is shot at an initial speed of,${\mathit{v}}_{\mathbf{0}}\mathbf{=}\mathbf{ }\mathbf{2}\mathbf{.00}\mathbf{ }\mathbf{\times }\mathbf{ }{\mathbf{10}}^{\mathbf{6}}\mathbf{ }\mathbf{\text{m/s}}$ at angle$\mathit{\theta }\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{40}\mathbf{.0}\mathbf{°}$  from an x axis. It moves through a uniform electric field. A screen for detecting electrons is positioned parallel to the y axis, at distance$\mathit{x}\mathbf{ }\mathbf{=}\mathbf{3}\mathbf{.00}\mathbf{ }\mathbf{\text{m}}$. In unit-vector notation, what is the velocity of the electron when it hits the screen?

A uniform electric field exists in a region between two oppositely charged plates. An electron is released from rest at the surface of the negatively charged plate and strikes the surface of the opposite plate, 2.0 cm  away, in a time$1.5\times {10}^{-8}s.$. (a) What is the speed of the electron as it strikes the second plate? (b) What is the magnitude of the electric field?

An electric dipole consists of charges +2e and -2e separated by 0.78 nm. It is in an electric field of strength$3.4\times {10}^6 N/C$. Calculate the magnitude of the torque on the dipole when the dipole moment is (a) parallel to, (b) perpendicular to, and (c) antiparallel to the electric field.

An electric dipole consisting of charges of magnitude$\mathbf{1}\mathbf{.50}\mathbf{ }\mathbf{\text{nC}}$  separated by$\mathbf{6}\mathbf{.20}\mathbf{ }\mathbf{\text{mm}}$ .  is in an electric field of strength $\mathbf{1100}\mathbf{ }\mathbf{\text{N/C}}$.What are (a) the magnitude of the electric dipole moment and (b) the difference between the potential energies for dipole orientations parallel and anti-parallel to $\overrightarrow{\mathbf{E}}$? :

A certain electric dipole is placed in a uniform electric field  $\overrightarrow{\mathbf{E}}$ of magnitude.$\mathbf{20}\mathbf{ }\mathbf{\text{N/C}}$ Figure 22-62 gives the potential energy of the dipole versus the angle u between E and the dipole moment  $\overrightarrow{\mathbf{p}}$. The vertical axis scale is set by${\mathit{U}}_{\mathbf{s}}\mathbf{=}\mathbf{ }\mathbf{1}\mathbf{.00}\mathbf{ }\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{28}}\mathbf{\text{J}}$ .What is the magnitude of $\overrightarrow{\mathbf{p}}$ ?

How much work is required to turn an electric dipole  $\mathbf{180}\mathbf{°}$in a uniform electric field of magnitude$\mathit{E}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{46}\mathbf{.0}\mathbf{ }\mathbf{\text{N/C}}\mathbf{ }$ if the dipole moment has a magnitude of $\mathit{p}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{3}\mathbf{.02}\mathbf{ }\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{25}}\mathbf{\text{C.m}}$and the initial angle is $\mathbf{64}\mathbf{°}$?

A certain electric dipole is placed in a uniform electric field of magnitude $\mathbf{40}\mathbf{ }\mathbf{\text{N/C}}$

. Figure 22-63 gives the magnitude t of the torque on the dipole versus the angle  between field and the dipole moment .The vertical axis scale is set by${\mathit{t}}_{\mathbf{s}}\mathbf{=}\mathbf{ }\mathbf{1}\mathbf{.00}\mathbf{ }\mathbf{\times }\mathbf{ }{\mathbf{10}}^{\mathbf{-}\mathbf{28}}\mathbf{\text{N.m}}$. What is the magnitude of  $\overrightarrow{\mathbf{p}}$?

Find an expression for the oscillation frequency of an electric dipole of dipole moment $\overrightarrow{\mathbf{p}}$ and rotational inertia I for small amplitudes of oscillation about its equilibrium position in a uniform electric field of magnitude E.

(a) what is the magnitude of an electron’s acceleration in a uniform electric field of magnitude $\mathbf{1}\mathbf{.40}\mathbf{ }\mathbf{\times }{\mathbf{10}}^{\mathbf{6}}\mathbf{\text{N/C}}$? (b) How long would the electron take, starting from rest, to attain one-tenth the speed of light? (c) How far would it travel in that time?

Three particles, each with positive chargeQ, form an equilateral triangle, with each side of length d .What is the magnitude of the electric field produced by the particles at the midpoint of any side?

In Fig. 22-64a, a particle of charge$\mathbf{+}\mathit{Q}$ produces an electric field of magnitude ${\mathit{E}}_{\mathbf{p}\mathbf{a}\mathbf{r}\mathbf{t}}$at point P, at distance R from the particle. In Fig. 22-64b, that same amount of charge is spread uniformly along a circular arc that has radius R and subtends an angle$\theta$. The charge on the arc produces an electric field of magnitude  at its center of curvatureP. For what value of does the electric field${\mathit{E}}_{\mathbf{a}\mathbf{r}\mathbf{c}}\mathbf{ }\mathbf{=}\mathbf{0}\mathbf{.500}{\mathit{E}}_{\mathbf{p}\mathbf{a}\mathbf{r}\mathbf{t}}$? (Hint: You will probably resort to a graphical solution.)

A proton and an electron form two corners of an equilateral triangle of side length $\mathbf{2}\mathbf{.0}\mathbf{ }\mathbf{\times }\mathbf{ }{\mathbf{10}}^{\mathbf{-}\mathbf{6}}\mathit{m}$.What is the magnitude of the net electric field these two particles produce at the third corner?

A charge (uniform linear density$\mathbf{=}\mathbf{ }\mathbf{9}\mathbf{.0}\mathbf{ }\mathbf{\text{nC/m}}$) lies on a string that is stretched along an x axis from $\mathit{x}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{0}$to $\mathit{x}\mathbf{ }\mathbf{=}\mathbf{3}\mathbf{.0}\mathbf{ }\mathbf{\text{m}}$. Determine the magnitude of the electric field at $\mathit{x}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{4}\mathbf{.0}\mathbf{ }\mathbf{\text{m}}$ on the x axis.

In Fig. 22-65, eight particles form a square in which distance$\mathit{d}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{2}\mathbf{.0}\mathbf{ }\mathbf{\text{cm}}$. The charges are,${\mathit{q}}_{\mathbf{1}}\mathbf{=}\mathbf{+}\mathbf{3}\mathit{e}$, ${\mathit{q}}_{\mathbf{2}}\mathbf{=}\mathbf{+}\mathit{e}$, ${\mathit{q}}_{\mathbf{3}}\mathbf{=}\mathbf{-}\mathbf{5}\mathit{e}$,${\mathit{q}}_{\mathbf{4}}\mathbf{=}\mathbf{-}\mathbf{2}\mathit{e}$ , ${\mathit{q}}_{\mathbf{5}}\mathbf{=}\mathbf{+}\mathbf{3}\mathit{e}$, ${\mathit{q}}_{\mathbf{6}}\mathbf{=}\mathbf{+}\mathit{e}$,${\mathit{q}}_{\mathbf{7}}\mathbf{=}\mathbf{-}\mathbf{5}\mathit{e}$ and ${\mathit{q}}_{\mathbf{8}}\mathbf{=}\mathbf{+}\mathit{e}$. In unit-vector notation, what is the net electric field at the square’s center?

Two particles, each with a charge of magnitude$\mathbf{12}\mathbf{ }\mathbf{\text{nC}}$, are at two of the vertices of an equilateral triangle with edge length$\mathbf{2}\mathbf{.0}\mathbf{ }\mathbf{\text{m}}$. What is the magnitude of the electric field at the third vertex if (a) both charges are positive and (b) one charge is positive and the other is negative?

Humid air brakes down (its molecules become ionized) in an electric field of $\mathbf{3}\mathbf{.0}\mathbf{ }\mathbf{\times }{\mathbf{10}}^{\mathbf{6}}\mathbf{\text{\hspace{0.17em}N/C}}$. In that field, what is the magnitude of the electrostatic force on (a) an electron and (b) an ion with a single electron missing?

A charge of  $\mathbf{20}\mathbf{ }\mathbf{\text{nC}}$is uniformly distributed along a straight rod of length $\mathbf{4}\mathbf{.0}\mathbf{ }\mathbf{\text{m}}$ that is bent into a circular arc with a radius of $\mathbf{2}\mathbf{.0}\mathbf{ }\mathbf{\text{m}}$ .What is the magnitude of the electric field at the center of curvature of the arc?

An electron is constrained to the central axis of the ring of charge of radius R in Fig. 22-11, with.$\mathit{z}\mathbf{ }\mathbf{\ll }\mathbf{ }\mathit{R}$ Show that the electrostatic force on the electron can cause it to oscillate through the ring center with an angular frequency$\mathit{\omega }\mathbf{=}\sqrt{\frac{\mathbf{e}\mathbf{q}}{\mathbf{4}\mathbf{\pi }{\mathbf{\in }}_{\mathbf{o}}\mathbf{m}{\mathbf{R}}^{\mathbf{3}}}}$where, q is the ring’s charge and m is the electron’s mass.

The electric field in ax-y plane produced by a positively charged particle is $\mathbf{7}\mathbf{.2}\mathbf{(}\mathbf{4}\mathbf{.0}\widehat{\mathbf{i}}\mathbf{+}\mathbf{ }\mathbf{3}\mathbf{.0}\widehat{\mathbf{j}}\mathbf{)}\mathbf{ }\mathbf{\text{N/C}}$ at the point (3.0, 3.0) cm and  $\mathbf{100}\widehat{\mathbf{i}\mathbf{ }}\mathbf{\text{N/C}}$at the point (2.0, 0) cm. What are the (a) x and (b) y coordinates of the particle? (c) What is the charge of the particle?

(a) What total (excess) charge q must the disk in Fig. 22-15 have for the electric field on the surface of the disk at its center to have magnitude $\mathbf{3}\mathbf{.0}\mathbf{ }\mathbf{\times }\mathbf{ }{\mathbf{10}}^{\mathbf{6}}\mathbf{\text{N/C}}$, the E value at which air breaks down electrically, producing sparks? Take the disk radius as.$\mathbf{2}\mathbf{.5}\mathbf{\text{cm}}$ (b) Suppose each surface atom has an effective cross-sectional area of$\mathbf{0}\mathbf{.015}\mathbf{ }{\mathbf{\text{nm}}}^{\mathbf{\text{2}}}$. How many atoms are needed to make up the disk surface? (c) The charge calculated in (a) results from some of the surface atoms having one excess electron. What fraction of these atoms must be so charged?

In Fig. 22-66, particle 1 (of charge$\mathbf{+}\mathbf{1}\mathbf{.00}\mathbf{ }\mathbf{\text{μC}}$), particle 2 (of charge), and particle 3 (of charge Q) form an equilateral triangle of edge length a. For what value of Q (both sign and magnitude) does the net electric field produced by the particles at the center of the triangle vanish?

In Fig. 22-67, an electric dipole swings from an initial orientation i (${\mathit{\theta }}_{\mathbf{i}}\mathbf{ }\mathbf{=}\mathbf{20}\mathbf{.0}\mathbf{°}$) to a final orientation f ($\mathbf{⟨}\mathbf{f}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{20}\mathbf{.0}\mathbf{°}$) in a uniform external electric field $\overrightarrow{\mathbf{E}}$ . The electric dipole moment is$\mathbf{1}\mathbf{.60}\mathbf{ }\mathbf{\times }\mathbf{ }{\mathbf{10}}^{\mathbf{-}\mathbf{27}}\mathbf{\text{C.m}}$; the field magnitude is$\mathbf{3}\mathbf{.00}\mathbf{ }\mathbf{\times }\mathbf{ }{\mathbf{10}}^{\mathbf{6}}\mathbf{\text{N/C}}$. What is the change in the dipole’s potential energy?

A particle of charge $\mathbf{-}{\mathit{q}}_{\mathbf{1}}$ is at the origin of an x axis. (a) At what location on the axis should a particle of charge $\mathbf{-}\mathbf{4}{\mathit{q}}_{\mathbf{1}}$  be placed so that the net electric field is zero at  $\mathit{x}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{2}\mathbf{.0}\mathbf{ }\mathit{m}\mathit{m}$on the axis? (b) If, instead, a particle of charge$\mathbf{+}\mathbf{4}{\mathit{q}}_{\mathbf{1}}$  is placed at that location, what is the direction (relative to the positive direction of the x axis) of the net electric field at $\mathit{x}\mathbf{=}\mathbf{2}\mathbf{.0}\mathbf{ }\mathit{m}\mathit{m}$?

Two particles, each of positive charge q, are fixed in place on a y axis, one at $\mathit{y}\mathbf{ }\mathbf{=}\mathbf{ }\mathit{d}$and the other at.$\mathit{y}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{-}\mathit{d}$ (a) Write an expression that gives the magnitude E of the net electric field at points on the x axis given by.$\mathit{x}\mathbf{ }\mathbf{=}\mathbf{ }\mathit{a}\mathit{d}$ (b) Graph E versus $\mathit{\alpha }$ for the range$\mathbf{0}\mathbf{<}\mathit{\alpha }\mathbf{<}\mathbf{ }\mathbf{4}$. From the graph, determine the values of $\mathit{\alpha }$ that give (c) the maximum value of E and (d) half the maximum value of E.

A clock face has negative point charges,$\mathbf{-}\mathit{q}$ , $\mathbf{-}\mathbf{2}\mathit{q}$, $\mathbf{-}\mathbf{3}\mathit{q}$. . . ,$\mathbf{-}\mathbf{12}\mathit{q}$fixed at the positions of the corresponding numerals. The clock hands do not perturb the net field due to the point charges. At what time does the hour hand point in the same direction as the electric field vector at the center of the dial? (Hint: Use symmetry.)

Calculate the electric dipole moment of an electron and a proton $\mathbf{4}\mathbf{.30}\mathbf{\text{ nm}}$ apart.

An electric field,$\overrightarrow{\mathbf{E}}$with an average magnitude of about $\mathbf{150}\frac{\mathbf{\text{N}}}{\mathbf{\text{C}}}$points downward in the atmosphere near Earth’s surface.We wish to “float” a sulfur sphere weighing$\mathbf{4}\mathbf{.4}\mathbf{\text{ N}}$  in this field by charging the sphere. (a) What charge (both sign and magnitude) must be used? (b) Why is the experiment impractical?

A circular rod has a radius of curvature $\mathbf{R}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{9}\mathbf{.00}\mathbf{\text{ cm}}$ and a uniformly distributed positive charge $\mathbf{Q}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{6}\mathbf{.25}\mathbf{\text{ pC}}$ and subtends an angle $\mathbf{\theta }\mathbf{=}\mathbf{ }\mathbf{2}\mathbf{.40}\mathbf{\text{\hspace{0.17em}rad}}$.What is the magnitude of the electric field that $\mathbf{\text{Q}}$produces at the center of curvature?

An electric dipole with dipole moment $\overrightarrow{\mathbf{p}}\mathbf{=}\mathbf{(}\mathbf{3}\mathbf{.00}\widehat{\mathbf{i}}\mathbf{+}\mathbf{4}\mathbf{.00}\widehat{\mathbf{j}}\mathbf{)}\mathbf{(}\mathbf{1}\mathbf{.24}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{30}\mathbf{ }}\mathbf{ }\mathbf{\text{C.m}}\mathbf{)}$is in an electric field$\overrightarrow{\mathbf{E}}\mathbf{=}\mathbf{\left(}\mathbf{4000}\mathbf{\text{ N/C}}\mathbf{\right)}\widehat{\mathbf{i}}$ (a) What is the potentialenergy of the electric dipole? (b) What is the torque acting on it?(c) If an external agent turns the dipole until its electric dipole moment is$\overrightarrow{\mathbf{p}}\mathbf{=}\mathbf{(}\mathbf{-}\mathbf{4}\mathbf{.00}\widehat{\mathbf{i}}\mathbf{+}\mathbf{3}\mathbf{.00}\widehat{\mathbf{j}}\mathbf{)}\mathbf{(}\mathbf{1}\mathbf{.24}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{30}\mathbf{ }}\mathbf{\text{ C.m}}\mathbf{)}$ how much work is done by the agent?

In Fig. 22-68, a uniform, upward electric field of magnitude  has been set up between two horizontal plates by charging the lower plate positively and the upper plate negatively. The plates have length $\mathit{L}\mathbf{ }\mathbf{=}\mathbf{10}\mathbf{.0}\mathbf{\text{ cm}}$ and separation$\mathit{d}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{2}\mathbf{.00}\mathbf{\text{ cm}}$. An electron is then shot between the plates from the left edge of the lower plate. The initial velocity $\overrightarrow{{\mathbf{V}}_{\mathbf{0}}}$ of the electron makes an angle$\mathit{\theta }\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{45}\mathbf{.0}\mathbf{°}$  with the lower plate and has a magnitude of $\mathbf{6}\mathbf{.00}\mathbf{ }\mathbf{\times }{\mathbf{10}}^{\mathbf{6}}\mathbf{\text{ m/s}}$. 

(a) Will the electron strike one of the plates? 

(b) If so, which plate and how far horizontally from the left edge will the electron strike?

For the data of Problem 70, assume that the charge q on the drop is given by $\mathit{q}\mathbf{ }\mathbf{=}\mathit{n}\mathit{e}$, where n is an integer and e is the elementary charge. 

(a) Find for each given value of q. 

(b) Do a linear regression fit of the values of versus the values of n and then use that fit to find e.

In Fig. 22-66, particle 1 (of charge $\mathbf{+}\mathbf{2}\mathbf{.00}\mathbf{\text{ pC}}$), particle 2 (of charge$\mathbf{-}\mathbf{2}\mathbf{.00}\mathbf{\text{ pC}}$), and particle 3 (of charge$\mathbf{+}\mathbf{5}\mathbf{.00}\mathbf{\text{ pC}}$) form an equilateral triangle of edge length $\mathit{a}\mathbf{ }\mathbf{=}\mathbf{9}\mathbf{.50}\mathbf{\text{ cm}}$. 

(a) Relative to the positive direction of the x-axisdetermines the direction of the force $\overrightarrow{{\mathbf{F}}_{\mathbf{3}}}$ on particle 3 due to the other particles by sketchingelectric field lines of the other particles.

 (b) Calculate the magnitude of $\overrightarrow{{\mathbf{F}}_{\mathbf{3}}}$

In Fig. 22-69, particle 1 of charge  ${\mathit{q}}_{\mathbf{1}}\mathbf{=}\mathbf{1}\mathbf{.00}\mathbf{\text{ pC}}$and particle 2 of charge ${\mathit{q}}_{\mathbf{2}}\mathbf{=}\mathbf{2}\mathbf{.00}\mathbf{\text{ pC}}$ are fixed at a distance  $\mathit{d}\mathbf{ }\mathbf{=}\mathbf{5}\mathbf{.00}\mathbf{ }\mathit{c}\mathit{m}$apart. In unit-vector notation, what is the net electric field at points

(a) A,

(b) B, and 

(c) C? 

(d) Sketch the electric field lines.