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Share Report Abuse Next Blog Create Blog Sign In The Heavenly Messenger Tuesday, January 27, 2009 Mastering Physics Solutions Use Wisely [ Item View ] The Electric Field of a Ball of Uniform Charge Density A solid ball of radius r_b has a uniform charge density rho . Part A What is the magnitude of the electric field E(r) at a distance r_b" title="" v:shapes="_x0000_i1028" align="middle" border="0" height="16" width="38">from the center of the ball? Express your answer in terms of rho , r_b , r , and epsilon_0 . The Electric Field of a Ball of Uniform Charge Density Part A What is the magnitude of the electric field E(r) at a distance r_b" title="" v:shapes="_x0000_i1484" align="middle" border="0" height="16" width="38">from the center of the ball? Hint 1. Gauss's law Gauss's law can be written as \oint \vec{E} \cdot d \vec{A} = \frac{q_{\rm encl}}{\epsilon_0 } , where d\vec A refers to an infinitesimal element of an imaginary Gaussian surface, q_encl is the net charge enclosed in the Gaussian surface, and epsilon_0 is the permittivity of free space. Always choose a Gaussian surface that matches the symmetry of the problem. Implicit in the question "what is E(r) ?" is the assumption that the electric field depends only on distance from the origin, which is also the center of the charged ball. Also, the ball has uniform charge density. Therefore, the electric field must point either radially outward or radially inward, since by symmetry there is no possibility for the electric field to point in any other direction. Given the symmetry of this problem, the best Gaussian surface to use is a sphere centered at the origin. Since the electric field is the same at all points on this surface, the constant E(r) can be "pulled out" of the integrand. The left side of Gauss's law reduces to E(r)A(r) , where A(r) is the surface area of a sphere with radius r . Hint 2. Find q_encl The q_encl in Gauss's law refers to the net charge enclosed inside the Gaussian surface. What is q_encl here? Hint 1. What is the volume of the sphere? If a body has uniform charge density rho , the charge in a volume V is \rho V (this formula is the same as that for the mass of a sphere of uniform mass density). What is the volume of a sphere with radius r ? Express your answer in terms of pi and r . V = (4/3)*pi*r^3 Express your answer in terms of rho , pi , and r_b . q_encl = (4/3)*pi*r_b^3*rho Express your answer in terms of rho , r_b , r , and epsilon_0 . E(r) = {\rho}{\cdot}\frac{\left(r_{b}{^{3}}\right)}{3{\cdot}epsilon_{0}{\cdot}r^{2}} Correct my answers The Heavenly Messenger: Mastering Physics Solutions Use Wisely http://rocketscient1st.blogspot.com/2009/01/mastering-physics-solutions-...... View Full Document