25. Electric Potential

What is the electric potential 3.0 cm away from a $-12\text{nC}$ point charge?

A $-9.0\text{nC}$ charge sits at the origin, and a $+18\text{nC}$ charge is at $x=2.0\mathrm{cm}$. At what location(s) is the potential zero?

A uniform electric field with strength $12\mathrm{V}/\mathrm{m}$ points from point A to point B, which are on the x-axis: $x_{\text{A}}=45\mathrm{m}$ and $x_{\mathrm{B}}=12\mathrm{m}$. What is the potential difference $V_{\text{A}}-V_{\mathrm{B}}$ ?

A spherical balloon is charged so that the potential on its surface is $40\mathrm{V}$. The balloon is now deflated to half its original radius (with no loss of charge). What is the new potential on its surface?

Metal sphere A is charged until it has a potential of $200\mathrm{V}$. Metal sphere B is initially uncharged, and has a radius twice the radius of sphere A. A small wire is used to connect spheres A and B, so that they form one continuous equipotential surface. What is the final potential on the surface of sphere B?

What are the strength and direction of the electric field 1.0 mm from (a) a proton and

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. (a) 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?

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?

(a) 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? (b) What is the potential of the sphere's surface relative to infinity?

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. (c) 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?

Two large, parallel conducting plates carrying op­posite charges of equal magnitude are separated by 2.20 cm. (a) If the surface charge density for each plate has magnitude 47.0 nC/m^2, what is the magnitude of E in the region between the plates?

An infinitely long line of charge has linear charge den­sity 5.00x10^-12 C/m. A proton (mass 1.67x10^-27 kg, charge +1.60x10^-19 C) is 18.0 cm from the line and moving directly toward the line at 3.50x10^3 m/s. (a) Calculate the proton's initial kinetic energy.

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

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

An electron is to be accelerated from 3.00x10^6 m/s to 8.00x10^6 m/s. (a) Through what potential difference must the electron pass to accomplish this?

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

A positive charge q is fixed at the point x = 0, y = 0, and a negative charge -2q is fixed at the point x = a, y = 0. (a) Show the positions of the charges in a diagram.

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. E23.19). Take the electric potential to be zero at infinity. Find (b) the potential at point B.

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. E23.19). Take the electric potential to be zero at infinity. Find (a) the potential at point A.

Two point charges of equal magnitude Q are held a distance d apart. Consider only points on the line passing through both charges. (a) 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 of equal magnitude Q are held a distance d apart. Consider only points on the line passing through both charges. (a) 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?).

Point charges q_1 = +2.00 μC and q_2 = -2.00 μC 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 bis at the empty corner closest to 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?

A small particle has charge -5.00 μC and mass 2.00x10^-4 kg. It moves from point A, where the electric po-tential 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.

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

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

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?

An infinitely long line of charge has linear charge den­sity 5.00x10^-12 C/m. A proton (mass 1.67x10^-27 kg, charge +1.60x10^-19 C) is 18.0 cm from the line and moving directly toward the line at 3.50x10^3 m/s. (b) How close does the proton get to the line of cha rge?

A thin rod of length L and total charge Q has the nonuniform linear charge distribution λ(x)=λ ₀x/L, where x is measured from the rod's left end. b. What is the electric potential on the axis at distance d left of the rod's left end?

Two positive point charges are 5.0 cm apart. If the electric potential energy is 72 μJ, what is the magnitude of the force between the two charges?

A 1.0-mm-diameter ball bearing has 2.0×10^9 excess electrons. What is the ball bearing's potential?

CALC The electric field in a region of space is Ex=5000x V/m , where x is in meters. b. Find an expression for the potential V at position x. As a reference, let V=0 V at the origin.

CALC An electric dipole at the origin consists of two charges ±q spaced distance s apart along the y-axis. a. Find an expression for the potential V(x, y) at an arbitrary point in the xy-plane. Your answer will be in terms of q, s, x, and y.

INT CALC Two positive point charges q are located on the y-axis at y = ±a. a. Write an expression for the electric potential at position x on the x-axis.