**Problem 1: UP 6.46**A total charge \( Q \) is distributed uniformly throughout a spherical shell with inner and outer radii \( r_1 \) and \( r_2 \), respectively. Show that the electric field due to the charge is \[\mathbf{E}(r) = \begin{cases} 0, & r \leq r_1 \\ \frac{Q}{4 \pi \varepsilon_0 r^2} \left( \frac{r^3 - r_1^3}{r_2^3 - r_1^3} \right) \hat{r}, & r_1 \leq r \leq r_2 \\ \frac{1}{4 \pi \varepsilon_0 r^2} \, Q \, \hat{r}, & r \geq r_2 \end{cases}\]In this expression:- \( \varepsilon_0 \) represents the permittivity of free space.- \( \hat{r} \) denotes the radial unit vector.The piecewise function describes the electric field \( \mathbf{E}(r) \) in three regions:1. Inside the inner radius \( r_1 \), where no electric field exists.2. Between radii \( r_1 \) and \( r_2 \), where the electric field depends on the position \( r \) and the distribution of charge.3. Outside the outer radius \( r_2 \), where the field behaves as if all the charge \( Q \) were concentrated at the center.This setup is a classic example of applying Gauss's law to spherically symmetric charge distributions.

Answered: Image /qna-images/answer/3418a2ad-ded7-4e4f-9a06-77177dc5690d.jpg

Problem 1: UP 6.46 A total charge Q is distributed uniformly throughout a spherical shell of inner and outer radii ₁ and 12, respectively. Show that the electric field due to the charge is r≤r₁ 1₁ ≤r ≤r₂ 0, Q E(r)=4neor²r²³ - r³, 1 Q 4περ 12 Î, f, r≥ 12

Problem 1: UP 6.46 A total charge Q is distributed uniformly throughout a spherical shell of inner and outer radii ₁ and 12, respectively. Show that the electric field due to the charge is r≤r₁ 1₁ ≤r ≤r₂ 0, Q E(r)=4neor²r²³ - r³, 1 Q 4περ 12 Î, f, r≥ 12

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

See similar textbooks

Similar questions

Two infinite nonconducting plates are parallel to each other, with a distance d = 12.0 cm between them. Each plate carries a uniform charge distribution with the surface charge density o = 4.28 μ C/m². What is the magnitude of the electric field at point R (with r = 17.0 cm)? ச 16 R 22 0.00 N/C 4.84 x 105 N/C O 5.80 x 105 N/C 3.87 x 105 N/C 3.39 X 105 N/C X
Three nonconducting strips are bent to form arcs and, when assembled, they form part of a circle of radius r-6.47 cm. The three strips have linear charge densities of A₁-88.0 nC/m, 1₂-179 nC/m, and 23-268 nC/m, respectively, and subtend angles of 60°, 120°, and 45°, respectively, at the center. 120 λ₁ 2₂ (a) Determine the electric potential at the center of the circle of which the strips form a part. V (b) You use a fourth nonconducting strip to close the circle. What should be the linear charge density on this strip if the potential at the center of the circle is to be zero? nC/m
The figure shows a closed Gaussian surface in the shape of a cube of edge length 2.50 m. It lies in a region where the electric field is given by E = (1.60x +3.08)i +7.77 +5.70 kN/C, with x in meters. What is the net charge contained by the cube? Number i 1.2e-10 Units C x
+X, to positive infinity. The line carries positive charge with a uniform linear charge density Ao. A continuous line of charge lies along the x axis, extending from x = (a) What is the magnitude of the electric field at the origin? (Use the following as necessary: 1o, xo, and ke.) 0. %3D (b) What is the direction of the electric field at the origin? O +X +y O -z
A thin, flat ring with inner radius R1 and outer radius R2 is uniformly charged on one side with a surface charge density σ > 0. a. Calculate the magnitude of the electric field at P, in newtons per coulomb, for the parameter values R1 = 0.086 m, R2 = 0.23 m, σ = 0.24 × 10-4 C/m2, and h = 0.15 m.
The figure shows a hollow sphere carrying a total charge of +Q which is distributed uniformly throughout the volume. a is the sphere's inner radius, while b is its outer radius. Find the electric field's magnitude in the region where b>r>a or a
In the figure shown in E, and E, be magnitude of electric field at A and B due to the hemisphere ofradius R and uniform surface charge density ở then which ofthe following is correct. ++ xx (a) E,-E, = , (b) E+E %3D %3D 4 E0 4. + B R メxxx 2R (c) E,-E,=- (d) E, = Eg 4 €, %3D 2R メ

a wire of length L=9.6 m, lies a long the x axis with its left end at the origin of th xy-plane as shown in the figure . The wire caries a linear charge density 160 nC/m. The magnitude of the X-components of the electric field in N/C at point :P(x=9.6,y=5.6) equals Hint: ſdx (b-x)/[{b-x) 2 +a²]1.5 =1/[(b- . x)2 +a²jò.5 %3D P. a
Figure (a) shows a nonconducting rod with a uniformly distributed charge +Q. The rod forms a 10/23 of circle with radius R and produces an electric field of magnitude Earc at its center of curvature P. If the arc is collapsed to a point at distance R from P (see Figure (b)), by what factor is the magnitude of the electric field at P multiplied? +Q +Q R (a) (b) Number i ! Units This answer has no units
QUESTION 2 2. Consider the following sphere inside a sphorical shell. Both are made of conducting material The radius of the sphere is R=1.65 cm. Inner radius and outer radii are 2R and 3R, respectively. Total charge of inner sphere is Q-1.91 nC. The charge of shell is -3Q. What is the magnitude of electric field on the outer surface of the sholl (in unit of N/C) ? ABCD QUESTION 3 3. Two paratiel plates of area 0.50 m² are each charged with uniformly with +0 and +20 as shown, where Q- 1.84 nC. Two plates are separated by 1 cm. (You may approximate the plates to be infinite planes.) What is the y-component of the electric field in region between the two plates (in unit of N/C)? +2Q +Q +y
A line of charge lying along the x-axis starts at x=+x0 and extends to positive infinity. It has a nonuniform linear charge density λ=8λ0x0/x, where λ0 is a positive constant. The magnitude of the electric field at the origin O is
y 0| X The positive charge Q is evenly distributed on the y-axis between y = 0 and y = a. Find the y- component of the electric field vector at a distance x from the origin on the positive x-axis.