In one line
Divergence of a vector field measures the net outward flow (flux) per unit volume from a point, showing whether the point acts as a source or a sink.
Explained simply
Imagine water flowing in a pipe. If at some point a tap is adding water, more water flows out of a small box around that point than flows in: positive divergence (a source). If there is a drain, more flows in than out: negative divergence (a sink). If water just passes through, inflow equals outflow: zero divergence.
How it works
- For \(\vec{A} = A_x\hat{i} + A_y\hat{j} + A_z\hat{k}\), divergence is the dot product of \(\nabla\) with \(\vec{A}\): \(\nabla\cdot\vec{A} = \dfrac{\partial A_x}{\partial x} + \dfrac{\partial A_y}{\partial y} + \dfrac{\partial A_z}{\partial z}\).
- The result is a scalar.
- Definition by flux: \(\nabla\cdot\vec{A} = \lim_{\Delta V\to0}\dfrac{\oint_S\vec{A}\cdot d\vec{S}}{\Delta V}\), the outward flux per unit volume.
- Positive: source; negative: sink; zero everywhere: the field is solenoidal (no sources or sinks).
- Physics link: \(\nabla\cdot\vec{E} = \rho/\varepsilon_0\) (charges are sources of \(\vec{E}\)) and \(\nabla\cdot\vec{B} = 0\) (no magnetic monopoles, so \(\vec{B}\) is solenoidal).
Key formula
\(\nabla\cdot\vec{A} = \dfrac{\partial A_x}{\partial x} + \dfrac{\partial A_y}{\partial y} + \dfrac{\partial A_z}{\partial z}\). \(A_x, A_y, A_z\): components of the vector field. The unit is that of \(A\) per metre.
Quick example
For \(\vec{A} = x^2\hat{i} + y^2\hat{j} + z^2\hat{k}\): \(\nabla\cdot\vec{A} = 2x + 2y + 2z\); at (1, 1, 1) it is 6, so that point acts as a source. For \(\vec{B} = y\hat{i} + z\hat{j} + x\hat{k}\): \(\nabla\cdot\vec{B} = 0 + 0 + 0 = 0\), so \(\vec{B}\) is solenoidal.
Common mistakes
- Writing the divergence as a vector; it is a scalar.
- Differentiating \(A_x\) with respect to \(y\) instead of \(x\).
- Confusing solenoidal (zero divergence) with irrotational (zero curl).
How MU asks it
Asked as "Explain the physical significance of divergence" (3 marks), "Show that the field is solenoidal" (3 marks), or in Maxwell's equations. Give the formula, flux-per-volume meaning, source/sink explanation, and a field-line sketch of positive, negative and zero divergence.