# Step-by-Step Solution ### **Given:** - The force field is \(\vec{F} = f(r) \vec{r}\) - For the case of **zero divergence**, show that the force magnitude is **inversely proportional to \(r^2\)**. --- ## **Step 1: Write the Divergence in Spherical Coordinates** For a **radial vector field** \(\vec{F} = f(r)\vec{r}\): - \(\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}\) - \(r = \sqrt{x^2 + y^2 + z^2}\) The divergence in spherical coordinates for a radial field: \[ \nabla \cdot \vec{F} = \frac{1}{r^2} \frac{d}{dr} \left( r^2 f(r) r \right) \] But \(\vec{F} = f(r)\vec{r}\), so the magnitude in the radial direction is \(f(r) r\). But actually, more generally, \[ \vec{F} = f(r) \vec{r} = f(r) r \hat{r} \] So the magnitude in the radial direction is \(F_r = f(r) r\). For a general radial vector field: \[ \nabla \cdot \vec{F} = \frac{1}{r^2} \frac{d}{dr} \left( r^2 F_r \right) \] where \(F_r = f(r) r\). --- ## **Step 2: Substitute \(F_r\) into the Divergence** \[ \nabla \cdot \vec{F} = \frac{1}{r^2} \frac{d}{dr} \left( r^2 f(r) r \right) = \frac{1}{r^2} \frac{d}{dr} \left( r^3 f(r) \right) \] --- ## **Step 3: Set Divergence to Zero** Given zero divergence: \[ \frac{1}{r^2} \frac{d}{dr} \left( r^3 f(r) \right) = \] \[ \frac{d}{dr} \left( r^3 f(r) \right) = \] \[ r^3 f(r) = C \] where \(C\) is a constant. --- ## **Step 4: Solve for \(f(r)\)** \[ f(r) = \frac{C}{r^3} \] --- ## **Step 5: Find the Magnitude of the Force Field** \[ |\vec{F}| = |f(r) \vec{r}| = |f(r)| |\vec{r}| = |f(r)| r \] \[ |\vec{F}| = \left| \frac{C}{r^3} \right| r = \frac{|C|}{r^2} \] --- ## **Final Answer** ### **For zero divergence, the force magnitude is inversely proportional to \(r^2\):** \[ \boxed{|\vec{F}| \propto \frac{1}{r^2}} \] --- ### **Summary Table** | Step | Equation | Result | |----------------|---------------------------------------------|--------------------------------| | Start | \(\vec{F} = f(r)\vec{r}\) | | | Divergence | \(\nabla \cdot \vec{F} = \frac{1}{r^2} \frac{d}{dr}(r^3 f(r))\) | | | Set to zero | \(\frac{d}{dr}(r^3 f(r)) = \) | \(r^3 f(r) = C\) | | Solve | \(f(r) = \frac{C}{r^3}\) | | | Magnitude | \(|\vec{F}| = |f(r)| r = \frac{|C|}{r^2}\) | \(|\vec{F}| \propto \frac{1}{r^2}\) | --- If you need the solution in a different format (like detailed written steps), let me know!