A disk of radius R = 4.0 cm is sprayed with a charged paint so that the radial charge density (the charge per area in a ring of radius r) varies continually with radial distance r from the center in the following manner: σ = −(5.0 C/m²)r R Find the potential (in GV), with respect to zero potential at infinity, at a point 3 cm above the center. × GV Additional Materials Reading
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- 8. Figure below shows a ring of outer radius R = 13.0 cm and inner radius l'inner = 0.200R. It has uniform surface charge density 0 = 6.20 pC/m². With V = 0 at infinity, find the electric potential at point P on the central axis of the ring, at distance z = 2.00R from the center of the ring. 6 dQ K √ ₁7 - Pl What is your dQ? What is your infinitesimal area element? (a) Start with the formula for the potential: V = k What are your vectors r and r'? What is the distance to point P? What is dV? Potential due to a small ring of charge on the disk? (b) Write out the integral that you need to compute to get V. What are the bounds? (c) Once you get an expression for V, solve numerically. (d) Check to see if the units of your expression makes sense for V.A uniformly charged insulating rod of length 12.0 cm is bent into the shape of a semicircle as shown in the figure below. The rod has a total charge of -9.00 μC. Find the electric potential at O, the center of the semicircle. x All the charge is equidistant from the center. Would the potential change if we gathered all of the charge into a single point at this same distance? MV ⓇThe electric potential on the surface of a charged conducting sphere is 210 V, and 10.0 cm farther from the center of the sphere the potential is 140 V. (a) Determine the radius of the sphere. HINT: Use a ratio of the two given potentials to solve for the radius. 20 cm (b) Determine the charge on the sphere. 4.67 nC The electric potential on the surface of another charged conducting sphere is 240 V, and 10.0 cm farther from the center the magnitude of the electric field is 410 V/m. (c) Determine all possible values for the radius of the sphere. (Enter your answers from smallest to largest. If only one value exists, enter "NONE" in the second answer blank.) HINT: Use the ratio of the potential at the surface to the E field at the surface plus 10.0 cm to solve for the radii. cm r2 = cm (d) Determine the charge on the sphere for each value of r. (If only one value exists, enter "NONE" in the second answer blank.) 91 = nC 92 =
- Point Charges 12 v.B If Q1 is a negative charge, what is the direction of the potential V1 at point P? Q1 O The potential has no direction. H Q2 13 Two point charges (Q1, Q2) and point P in empty space are located on the corners of a rectangle of length L = 3.0 m and height H = 1.6 m. V1= electric potential (or voltage) generated by the charge Q1 V2 = electric potential (or voltage) generated by the charge Q2 If the charges have values of Q1 = -3.5 µC and Q2 = +1.8 µC, what is the value of the total potential (or voltage) at point P? Enter the numerical value in Sl units. %3D %3D Type your answer...Consider a solid insulating sphere which has a total chargeof +3Q but is distributed as ρ(r) = βr, and has a radius of a. This issurrounded by a conducting shell that has a charge of −3Q placed onits outer surface. The inner radius is b and the outer radius is c. a) Determine β in terms of Q and a.b) Find the potential at all points in spaceProblem 2: A hollow cylindrical shell of length L and radius R has charge Q uniformly distributed along its length. What is the electric potential at the center of the cylinder? a) Compute the surface charge density n of the shell from its total charge and geometrical parameters. Vcenter = 1 Q 2 In 4л€ L t₂ S² b) Which charge dq is enclosed in a thin ring of width dz located at a distance z from the center of the cylinder (shown in Fig.2)? Which potential dV does this ring create at the center (you need to use the formula derived in the textbook for the potential of a charged ring along its axis). dz c) Sum up the contributions from all the rings along the cylinder by integrating dV with respect to z. Show that (The integral that you need to use here is d dt √²+a² R² + 1/2 + 1/1/20 √√R² +4-4 L² R2 L R FIG. 2: The scheme for Problem 2 [2 = ln(t + √₁² + a²) 1².) 2
- A body is charged with a uniform charge density p = 3.1 nC/m³. A spherical cavity is created inside the body with radius R = 0.58 m (all the charges that were inside the sphere were taken out of the body). Before the cavity was created the electric potential at point O (at the center of the cavity) was 40 = 102 V. What is the potential at O after creating the cavity?4. Figure below shows a ring of outer radius R = 13.0 cm, inner radius r= 0.200R, and uniform surface charge density o = 6.20 pC/m2. With V = 0 at infinity, find the electric potential at point P on the central axis of the ring, at distance z = 2.0OR from the center of the ring. %3D R