Question

Charges q1=−4Qq_1 = -4Q  and q2=+2Qq_2 = +2Q are located at 𝓍=−a𝓍 = -a and 𝓍=+a𝓍 = +a, respectively. What is the net electric flux through a sphere of radius 2a2a centered (a) at the origin and (b) at 𝓍=2a𝓍 = 2a?

Accepted Answer

Step 1: Recall Gauss's Law, which states that the net electric flux (Φ) through a closed surface is proportional to the total charge enclosed within the surface. Mathematically, it is expressed as: Φ = (q_{enclosed})/(ε_{0}), where q_{enclosed} is the total charge enclosed by the surface and ε_{0} is the permittivity of free space. Step 2: For part (a), the sphere of radius 2a is centered at the origin. Both charges q₁ = -4Q (at 𝓍 = -a) and q₂ = +2Q (at 𝓍 = +a) are inside the sphere. Calculate the total charge enclosed by summing the charges: q_{enclosed} = q₁ + q₂ = -4Q + 2Q. Substitute this value into Gauss's Law to find the net electric flux. Step 3: For part (b), the sphere of radius 2a is centered at 𝓍 = 2a. Determine whether the charges q₁ and q₂ are inside or outside the sphere. Note that q₁ is located at 𝓍 = -a, which is outside the sphere, and q₂ is located at 𝓍 = +a, which is also outside the sphere. Since no charges are enclosed by the sphere, q_{enclosed} = 0. Substitute this value into Gauss's Law to find the net electric flux. Step 4: For part (a), simplify the expression for the net electric flux using the value of q_{enclosed} calculated in Step 2. The flux will depend on the total enclosed charge and the permittivity of free space, ε_{0}. Write the final expression for the flux. Step 5: For part (b), since q_{enclosed} = 0, the net electric flux through the sphere is zero. This result follows directly from Gauss's Law, as no charge is enclosed within the sphere.

Charges $q_{1} = - 4 Q$ and $q sub 2 equals positive 2 Q$ are located at $𝓍 = - a$ and $𝓍 = + a$ , respectively. What is the net electric flux through a sphere of radius $2 a$ centered (a) at the origin and (b) at $script x equals 2 a$ ?

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Key Concepts

Electric Flux

Gauss's Law

Charge Distribution