Due wen 1. In each of the following regions, there is no charge. a) ID: The volt

Question

Due wen 1. In each of the following regions, there is no charge. a) ID: The voltage at x = o is 20 volts, and the voltage at x = 6 m is-10 volts. Find the voltages at x = 3 m and x = 4 m. The region here is x e [0m, 6m]. b) 2D: The voltage everywhere on the x = 0 m line is 3 volts. The voltage everywhere on the x -4 m line is -4 volts. The voltage everywhere on the y 0 m line is 7 volts, and the voltage everywhere on the y 6 m line is -1 volt. The region here is the rectangle bounded by the four lines described above. Determine the voltage at (x, y) (2m, 3m). e) 3D: Consider a rectangular box bounded by 6 rectangular planes. The voltage everywhere on the x 0 plane is 0 volts, but the voltage everywhere on thex -10m plane is 10 volts. The voltage everywhere on the y = 0 plane is 0 volts, but the voltage everywhere on the y 20 m plane is 13 volts. The voltage everywhere on the: -0 plane is 0 volts, but the voltage everywhere on the : - 15 m plane is -6 volts. Determine the voltage in the middle of the rectangular box. d) 3D, spherical: A region defined by a sphere of radius 3m has a constant voltage along its outside surface of 10 volts. What is the voltage at the center of the sphere? (Trick question?)

Explanation / Answer

given

a. 1D

V1 at x = 0 = 20 V

V2 at x = 6 = -10 V

so, V3 at x = 3 m

V1 - V3 = (V1 - V2)*(0 - 3)/(0 - 6) = 30/2 = 15 V

V3 = V1 - 15 = 5 V

similiarly at x = 4 m

V1 - V4 = (v1 - V2)*(0 - 4)/(0 - 6)

V1 - V4 = 20

V4 = 0 V

b. 2D

V at x = 0 = 3V

V at x = 4 = -4 V

hence voltage variation along x axis = -7V per 4 m = -1.75 V/m

V at y = 0 = 7 V

V at y = 6 = -1 V

hence voltager variation along y axis = -8 V per 6 m = -1.333 V/m

so voltage at (2,3) is Vt

voltage due to x variation at x = 2 is -1.75*2 = -3.5 V

voltage due to y variation at y = 3 is -1.333*3 = -4 V

so net voltage at (2,3) = -3.5-4 = -7.5 V

c. 3D

given

V at x = 0 plane is 0 V

V at y = 0 plane is 0 V

V at z = 0 plane is 0 V

V at x = 10 plane is 10 V

V at y = 20 plane is 13 V

V at z = 15 plane is -6 V

so x voltage variation = 10 V per 10 m = 1 V per m

y voltage variation = 13 V per 20 m => 0.65 V/m

z voltage variation = -6 V per 15 m = -0.4 V/m

so center of the cube is (5,10,7.5)

voltage at this point due to superpositions of the three variations of voltages are

V = 5*1 + 10*0.65 - 0.4*7.5 = 8.5 V

hence voltage at the center of the cuboid is 8.5 V

d. 3D spherical region, surface potential = 10 V

radius = 3m

now, as there is no charge enclosed ion thjis region, net electric field i 0

hence potneial remains constant inside this region

hence at the center of the sphere, voltage = 10 V = at surface of the sphere

Related Questions

Navigate

• Browse (All)
• Browse D
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.