Consider a sphere of radius R which carries
Consider a sphere of radius r which carries a uniform charge density ρ
Consider a sphere of radius R which carries a uniform charge density ρ. If a sphere of radius R/2 is carved out of it, as shown, the ratio EB/EA of magnitude of electric field EA and EB, respectively, at points A and B due to the remaining portion will be?
Solution: Consider a sphere of radius R which carries……….
Given : Radius sphere = R Radius of small sphere = R / 2
Uniform charge density of sphere = ρ.
EB / EA = ?
Charge inside small sphere, taken out from complete sphere,
![Q_1 = charge density (rho) * volume of small sphere = rho * [ {4 / 3} pi (R / 2)^3]](https://mywebpathshala.com/wp-content/plugins/wpmathpub/phpmathpublisher/img/math_971.5_797f1837e803f7dadb37b0b33658eb38.png "Q_1 = charge density (rho) * volume of small sphere = rho * [ {4 / 3} pi (R / 2)^3]")
Electric field at point A ( inside sphere) ,
(as total charge lying inside metallic complete sphere is zero so, E = 0. In a metallic sphere charge lies on surface only )
Electric field at point B, will be only due to smaller sphere as its inner surface have charge due to induction.
![{E}_(Electric field due to small sphere) = { 1 / in_0} * {[rho * 4 / 3 pi (R / 2)^3]} / {4 pi (R / 2)^2}](https://mywebpathshala.com/wp-content/plugins/wpmathpub/phpmathpublisher/img/math_943_6a8d7cb5282b96ddd543c44cd92d2afa.png "{E}_(Electric field due to small sphere) = { 1 / in_0} * {[rho * 4 / 3 pi (R / 2)^3]} / {4 pi (R / 2)^2}")
So, 
Electric field at point B (at surface of sphere) ,
![E_B = {1 / {4 pi in_0}} Q_2 / R^2 - {1 / {4 pi in_0}} * {1 / ({3R} / 2)^2} * [{rho * {4 pi} / 3 (R / 2)^3 }]](https://mywebpathshala.com/wp-content/plugins/wpmathpub/phpmathpublisher/img/math_944.5_72458bbf81028c149121b41915c7bb0a.png "E_B = {1 / {4 pi in_0}} Q_2 / R^2 - {1 / {4 pi in_0}} * {1 / ({3R} / 2)^2} * [{rho * {4 pi} / 3 (R / 2)^3 }]")
![E_B = {{1 / {4 pi in_0}} 1 / R^2 } * { [{rho * 4 / 3 pi R^3}] } - {1/ {4 pi in_0}} * {1 / ({3R} / 2)^2} * [{rho * {4 pi} / 3 (R / 2)^3 }]](https://mywebpathshala.com/wp-content/plugins/wpmathpub/phpmathpublisher/img/math_943.5_16593a8b045d4e7fd0cf23d4c1da9e1a.png "E_B = {{1 / {4 pi in_0}} 1 / R^2 } * { [{rho * 4 / 3 pi R^3}] } - {1/ {4 pi in_0}} * {1 / ({3R} / 2)^2} * [{rho * {4 pi} / 3 (R / 2)^3 }]")
![E_B = { rho R } / { in_0} [ 1 / 3 - 4 / {3 * 9 * 8}] = { rho R } / { 3 in_0} [ 1 - 1 / 18 ]](https://mywebpathshala.com/wp-content/plugins/wpmathpub/phpmathpublisher/img/math_970.5_e30bd72be336b64cdf849bd8279e320b.png "E_B = { rho R } / { in_0} [ 1 / 3 - 4 / {3 * 9 * 8}] = { rho R } / { 3 in_0} [ 1 - 1 / 18 ]")
Activities Supporting Speaking Skill Development
Consider a sphere of radius R which carries a uniform charge density ρ