A point charge $q_1=+2.40$ $\mu$C is held stationary at the origin. A second point charge $q_2=-4.30$ $\mu$C moves from the point $x=0.150$ m, $y=0$ to the point $x=0.250$ m, $y=0.250$ m. How much work is done by the electric force on $q_2$?

How much work would it take to push two protons very slowly from a separation of $2.00\times {10}^{-10}$ m (a typical atomic distance) to $3.00\times {10}^{-15}$ m (a typical nuclear distance)? If the protons are both released from rest at the closer distance in part (a), how fast are they moving when they reach their original separation?

A small metal sphere, carrying a net charge of $q_1=-2.80$ μC, is held in a stationary position by insulat­ing supports. A second small metal sphere, with a net charge of $q_2=-7.80$ μC and mass $1.50$ g, is projected toward $q_1$. When the two spheres are $0.800$ m apart, $q_2$, is moving toward $q_1$ with speed $22.0$ m/s (Fig. E$23.5$). Assume that the two spheres can be treated as point charges. You can ignore the force of gravity. What is the speed of $q_2$ when the spheres are $0.400$ m apart?

Two protons, starting several meters apart, are aimed directly at each other with speeds of $2.00\times {10}^5$ m/s, measured relative to the earth. Find the maximum electric force that these protons will exert on each other.

Two protons are released from rest when they are $0.750$ nm apart. What is the maximum acceleration they will achieve and when does this acceleration occur?

A small particle has charge $-5.00$ $\mu$C and mass $2.00\times {10}^{-4}$ kg. It moves from point $A$, where the electric potential is $V_A=+200$ V, to point $B$, where the electric potential is $V_B=+800$ V. The electric force is the only force acting on the particle. The particle has speed $5.00$ m/s at point $A$. What is its speed at point $B$? Is it moving faster or slower at $B$ than at $A$? Explain.

A particle with charge $+4.20$ nC is in a uniform electric field $E$ directed to the left. The charge is released from rest and moves to the left; after it has moved $6.00$ cm, its kinetic energy is $+2.20x{10}^{-6}$ J. What is the work done by the electric force?

Two stationary point charges $+3.00$ nC and $+2.00$ nC are separated by a distance of $50.0$ cm. An electron is released from rest at a point midway between the two charges and moves along the line connecting the two charges. What is the speed of the electron when it is $10.0$ cm from the $+3.00$-nC charge?

Point charges $q_1=+2.00$ $\mu$C and $q_2=-2.00$ $\mu$C are placed at adjacent corners of a square for which the length of each side is $3.00$ cm. Point $a$ is at the center of the square, and point $b$ is at the empty corner closest to $q_2$$q_2$. Take the electric potential to be zero at a distance far from both charges. (a) What is the electric potential at point a due to $q_1$ and $q_2$?

Two point charges of equal magnitude $Q$ are held a distance $d$ apart. Consider only points on the line passing through both charges. If the two charges have the same sign, find the location of all points (if there are any) at which (i) the potential (relative to infinity) is zero (is the electric field zero at these points?), and (ii) the electric field is zero (is the potential zero at these points?).

Two point charges $q_1=+2.40$ nC and $q_2=-6.50$ nC are $0.100$ m apart. Point $A$ is midway between them; point $B$ is $0.080$ m from $q_1$ and $0.060$ m from $q_2$ (Fig. E$23.19$). Take the electric potential to be zero at infinity. Find the potential at point $A$.

Two point charges $q_1 = +2.40$ nC and $q_2 = -6.50$ nC are $0.100$ m apart. Point $A$ is midway between them; point $B$ is $0.080$ m from $q_1$ and $0.060$ m from $q_2$ (Fig. E$23.19$). Take the electric potential to be zero at infinity. Find the potential at point $B$.

An electron is to be accelerated from $3.00\times {10}^6$ m/s to $8.00\times {10}^6$ m/s. Through what potential difference must the electron pass to accomplish this?

At a certain distance from a point charge, the potential and electric-field magnitude due to that charge are $4.98$ V and $16.2$ V/m, respectively. (Take $V = 0$ at infinity.) What is the distance to the point charge?

At a certain distance from a point charge, the potential and electric-field magnitude due to that charge are $4.98$ V and $16.2$ V/m, respectively. (Take $V=0$ at infinity.) What is the magnitude of the charge?

At a certain distance from a point charge, the potential and electric-field magnitude due to that charge are $4.98$ V and $16.2$ V/m, respectively. (Take $V = 0$ at infinity.) Is the electric field directed toward or away from the point charge?

A thin spherical shell with radius $R_1=3.00$ cm is concentric with a larger thin spherical shell with radius $R_2=5.00$ cm. Both shells are made of insulating material. The smaller shell has charge $q_1=+6.00$ nC distributed uniformly over its surface, and the larger shell has charge $q_2=-9.00$ nC distributed uniformly over its surface. Take the electric potential to be zero at an infinite distance from both shells. What is the electric potential due to the two shells at the following distance from their common center: (i) $r=0$; (ii) $r=4.00$ cm; (iii) $r=6.00$ cm?

An infinitely long line of charge has linear charge den­sity $5.00\times {10}^{-12}$ C/m. A proton (mass $1.67\times {10}^{-27}$ kg, charge $+1.60\times {10}^{-19}$ C) is $18.0$ cm from the line and moving directly toward the line at $3.50\times {10}^3$ m/s. Calculate the proton's initial kinetic energy.

An infinitely long line of charge has linear charge den­sity $5.00\times10^{-12}$ C/m. A proton (mass $1.67\times10^{-27}$ kg, charge $+1.60\times10^{-19}$ C) is $18.0$ cm from the line and moving directly toward the line at $3.50\times10^3$ m/s. How close does the proton get to the line of charge?

A very long insulating cylinder of charge of radius $2.50$ cm carries a uniform linear density of $15.0$ nC/m. If you put one probe of a voltmeter at the surface, how far from the surface must the other probe be placed so that the voltmeter reads $175$ V?

Two large, parallel conducting plates carrying op­posite charges of equal magnitude are separated by $2.20$ cm. If the surface charge density for each plate has magnitude $47.0$ nC/m², what is the magnitude of $E$ in the region between the plates?

Two large, parallel conducting plates carrying op­posite charges of equal magnitude are separated by $2.20$ cm. What is the potential difference between the two plates?

Two large, parallel conducting plates carrying op­posite charges of equal magnitude are separated by $2.20$ cm. The surface charge density for each plate has magnitude $47.0$ nC/m^2. If the separation between the plates is doubled while the surface charge density is kept constant at the given value, what happens to the magnitude of the electric field and to the po­tential difference?

Certain sharks can detect an electric field as weak as $1.0$ $\mu$V/m. To grasp how weak this field is, if you wanted to produce it between two parallel metal plates by connecting an ordinary $1.5$­V AA battery across these plates, how far apart would the plates have to be?

How much excess charge must be placed on a copper sphere $25.0$ cm in diameter so that the potential of its center, rela­tive to infinity, is $3.75$ kV? What is the potential of the sphere's surface relative to infinity?

A metal sphere with radius $r_a$ is supported on an insulating stand at the center of a hollow, metal, spherical shell with radius $r_b$. There is charge $+q$ on the inner sphere and charge $-q$ on the outer spherical shell. Calculate the potential $V\left(r\right)$ for (i) ; (ii) ; (iii) $r>r_b$. (Hint: The net potential is the sum of the potentials due to the individual spheres.) Take $V$ to be zero when $r$ is infinite.

A metal sphere with radius $r_a$ is supported on an insulating stand at the center of a hollow, metal, spherical shell with radius $r_b$. There is charge $+q$ on the inner sphere and charge $-q$ on the outer spherical shell. Show that the potential of the inner sphere with respect to the outer is $V_{a}b=q/\left(4\pi {ϵ}_0\right)(1/r_a-1/r_b)$.

A metal sphere with radius $r_a$ is supported on an insulating stand at the center of a hollow, metal, spherical shell with radius $r_b$. There is charge $+q$ on the inner sphere and charge $-q$ on the outer spherical shell. Use $E_r=-\partial V/\partial r=(-\partial /\partial r)\left(1/\right(4\pi {ϵ}_0\left)q/r\right)=\left[1/\right(4\pi {ϵ}_0\left)\right]\left(q/r^2\right)$ and the result from part (a) to show that the electric field at any point between the spheres has magnitude $E\left(r\right)=\left[V_{a}b/\right(1/r_a-1/r_b\left)\right]\left(1/r^2\right)$. Note: Part (a) asked to calculate the potential $V(r)$ for (i) $r < r_a$; (ii) $r_a < r < r_b$; (iii) $r > r_b$. (Hint: The net potential is the sum of the potentials due to the individual spheres.) Take $V$ to be zero when $r$ is infinite..

A metal sphere with radius $r_a$ is supported on an insulating stand at the center of a hollow, metal, spherical shell with radius $r_b$. There is charge $+q$ on the inner sphere and charge $-q$ on the outer spherical shell. Use $E_r=\left[1/\right(4\pi {ϵ}_0\left)\right]\left(q/r^2\right)$ and the result from part (a) to find the electric field at a point outside the larger sphere at a distance $r$ from the center, where $r>r_b$. Note: Part (a) asked to calculate the potential $V(r)$ for (i) $r < r_a$; (ii) $r_a < r < r_b$; (iii) $r>r_b$. (Hint: The net potential is the sum of the potentials due to the individual spheres.) Take $V$ to be zero when $r$ is infinite..

Suppose the charge on the outer sphere is not $-q$ but a negative charge of different magnitude, say $-Q$. Show that the answers for parts (b) and (c) are the same as before but the answer for part (d) is different. Note: Part (a) asked to calculate the potential $V(r)$ for (i) $r < r_a$; (ii) $r_a < r < r_b$; (iii) $r > r_b$. (Hint: The net potential is the sum of the potentials due to the individual spheres.) Take $V$ to be zero when $r$ is infinite. Part (b) asked to show that the potential of the inner sphere with respect to the outer is $V_ab=q/(4πϵ_0 ) (1/r_a -1/r_b)$. Part (c) asked to use $E_r=-∂V/∂r=(-∂/∂r) (1/(4πϵ_0 ) q/r)=[1/(4πϵ_0 )](q/r^2)$ and the result from part (a) to show that the electric field at any point between the spheres has magnitude $E(r)=[V_ab/(1/r_a -1/r_b )](1/r^2)$. Part (d) asked to use $E_r = [1/(4πϵ_0 )](q/r^2)$ and the result from part (a) to find the electric field at a point outside the larger sphere at a distance $r$ from the center, where $r > r_b$.