25. Electric Potential

A thin flat disk of radius R₀ carries a total charge Q that is distributed uniformly over its surface. The electric potential at a distance x on the x axis is given by V(x) = Q/ 2π∊₀R₀²[(x² + R²₀) ¹⸍² - x]. (See Example 23–10.) Show that the electric field at a distance x on the x axis is given by E(x) = Q/2π∊₀R₀² ( 1 - ( x / ( x² + R²₀))¹⸍². Make graphs of V(x) and E(x) as a function of x/R₀ for x/R₀ = 0 to 4. (Do the calculations in steps of 0.1.) Use Q = 5.0μC and R₀ = 10 cm for the calculation and graphs.

(II) The electric potential between two parallel conducting plates is given by V(x) = (8.5 V/m0 x + 4.5 V , with x = 0 taken at one of the plates and x positive in the direction toward the other plate. What is the charge density on the plates?

A thin rod of length L and total charge Q has the nonuniform linear charge distribution λ(x)=λ₀x/L, where x is measured from the rod's left end. What is the electric potential on the axis at distance d left of the rod's left end?

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?

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?

Two point charges qₐ and qᵦ are located on the x-axis at x=a and x=b. FIGURE EX25.36 is a graph of V, the electric potential. Draw a graph of Eₓ, the x-component of the electric field, as a function of x.

FIGURE P25.70 shows a thin rod of length L and charge Q. Find an expression for the electric potential a distance x away from the center of the rod on the axis of the rod.

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?

FIGURE EX26.12 is a graph of V versus x. Draw the corresponding graph of E_x versus x.

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. How close does the proton get to the line of charge?

A Van de Graaff generator (Fig. 23–58) can develop a very large potential difference, even millions of volts. Electrons are pulled off the belt by the high voltage pointed electrode (positive) at A, leaving the belt positively charged. (Recall Example 23–5 where we saw that near sharp points the electric field is high and ionization can occur.) The belt carries the positive charge up inside the spherical shell where electrons from the large conducting sphere are attracted over to the pointed conductor at B, leaving the outer surface of the conducting sphere positively charged. As more charge is brought up, the sphere reaches extremely high voltage. Consider a Van de Graaff generator with a sphere of radius 0.20 m. What is the electric potential on the surface of the sphere when electrical breakdown occurs ( E = 3 x 10⁶ V/m) ? Assume V = 0 at r = ∞.

The potential 1.0 cm from the surface of a metal sphere is 8000 V. The potential 3.0 cm from the surface is 4000 V. What is the radius of the sphere?

Two point charges, 3.4 μC and -2.0 μC, are placed 8.0 cm apart on the x axis. At what points along the x axis are

(a) the electric field zero and

(b) the potential zero? Let V = 0 at r = ∞.

A −1μC and a 5μC charge lie on a line, separated by 5cm. What is the electric potential halfway between the two charges?