Problem 54 - University Physics with Modern Physics

Question

A solid, nonconducting sphere of radius \(a\) has total charge \(Q\) and a uniform charge distribution. Using Gauss's Law, determine the electric field (as a vector) in the regions \(ra\) in terms of \(Q\).

Step-by-Step Solution

Answer

Answer: The electric field E(r) is as follows: - Inside the sphere (r < a): E(r) = (Q*r) / (4π*ε₀*a^3) * rˆ - Outside the sphere (r > a): E(r) = Q / (4π*ε₀*r^2) * rˆ

Step 1: Determine the Gaussian surface

Choose the Gaussian surface as a concentric sphere with radius r, where r is the distance from the center of the sphere. Using symmetry, it is evident that the electric field will have only a radial component.

Step 2: Applying Gauss's Law - Inside the sphere (r < a)

Let's consider the spherical Gaussian surface with radius r inside the sphere (r < a). As the electric field (E) and the area vector (dA) are both radially pointing outward (same direction), the angle between them (θ) is 0 degrees. Therefore, the dot product of E and dA in Gauss's Law is |E||dA|cosθ which simplifies to |E||dA|. To apply Gauss's Law, we need to determine the charge enclosed by the Gaussian surface of radius r. Since the sphere has a uniform charge density, we can find the charge density (ρ) as: ρ = Q / (4/3*π*a^3) The volume enclosed by the Gaussian surface is: V = (4/3)*π*r^3 Hence, the enclosed charge (q) is: q = ρ*V = Q*(r^3) / (a^3) Now, apply Gauss's Law to the Gaussian surface of radius r: ∮E⋅dA = q / ε₀ |E|∮dA = q / ε₀ |E|(4πr^2) = (Q*r^3)/(a^3*ε₀) Solving for E, we get for r < a: E = (Q*r)/(4π*ε₀*a^3)

Step 3: Applying Gauss's Law - Outside the sphere (r > a)

Now let's consider the spherical Gaussian surface with radius r outside the sphere (r > a). For this case, the angle (θ) between E and dA is still 0. The Gaussian surface encloses the whole sphere, hence enclosed charge (q) remains Q. Applying Gauss's Law to the Gaussian surface of radius r: ∮E⋅dA = q / ε₀ |E|∮dA = Q / ε₀ |E|(4πr^2) = Q/ε₀ Solving for E, we get for r > a: E = Q / (4π*ε₀*r^2)

Step 4: Writing the electric field as a vector

Since the electric field has only a radial component, we can write it in spherical coordinates as: E(r) = Er * rˆ = |E| * rˆ Combining our results for r < a and r > a, we get: Electric field E(r) = - For r < a: (Q*r) / (4π*ε₀*a^3) * rˆ - For r > a: Q / (4π*ε₀*r^2) * rˆ