Divergence and Central Force Field

Transcribed Lecture Notes (First 3 Pages)

Divergence, ∇⃗⋅\vec{\nabla} \cdot

We have seen gradient, that makes static scalar field into vector. But in case of divergence, we make a vector field, into some highly informative numbers. So that we can find the source, sink etc. You can consider a box, some fluids are coming into it and some are leaving. Divergence will tell you either the box is sink or source of the fluid.

Let's A⃗=Axe^x+Aye^y+Aze^z\vec{A} = A_x \hat{e}_x + A_y \hat{e}_y + A_z \hat{e}_z

Divergence by definition:

∇⃗⋅A⃗=(∂∂xe^x+∂∂ye^y+∂∂ze^z)⋅(Axe^x+Aye^y+Aze^z)\vec{\nabla} \cdot \vec{A} = \left(\frac{\partial}{\partial x}\hat{e}_x + \frac{\partial}{\partial y}\hat{e}_y + \frac{\partial}{\partial z}\hat{e}_z\right) \cdot \left(A_x \hat{e}_x + A_y \hat{e}_y + A_z \hat{e}_z\right)
=(∂Ax∂x+∂Ay∂y+∂Az∂z)= \left(\frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} + \frac{\partial A_z}{\partial z}\right)

Let's r⃗=xe^x+ye^y+ze^z\vec{r} = x\hat{e}_x + y\hat{e}_y + z\hat{e}_z. Find div r⃗\text{div } \vec{r}

r⃗=xe^x+ye^y+ze^z\vec{r} = x\hat{e}_x + y\hat{e}_y + z\hat{e}_z
∴∇⃗⋅r⃗=(∂∂xe^x+∂∂ye^y+∂∂ze^z)⋅(xe^x+ye^y+ze^z)\therefore \vec{\nabla} \cdot \vec{r} = \left(\frac{\partial}{\partial x}\hat{e}_x + \frac{\partial}{\partial y}\hat{e}_y + \frac{\partial}{\partial z}\hat{e}_z\right) \cdot \left(x\hat{e}_x + y\hat{e}_y + z\hat{e}_z\right)
=(∂x∂x+∂y∂y+∂z∂z)= \left(\frac{\partial x}{\partial x} + \frac{\partial y}{\partial y} + \frac{\partial z}{\partial z}\right)
=3= 3
∴∇⃗⋅r⃗=3\therefore \vec{\nabla} \cdot \vec{r} = 3

Find the divergence of central force field

Central force field:

A central force field is a spatial field where force exerted on an object is always directed towards or away from a single fixed point (center of force), magnitude depends only on object's distance from that point.

F⃗=f(r)r^\vec{F} = f(r)\hat{r}
  • F⃗\vec{F}→\rightarrow Force vector
  • f(r)f(r)→\rightarrow Scalar function that defines the magnitude of the force. It depends solely on scalar distance rr from the origin.
  • f(r)<0f(r) < 0→\rightarrow pulling toward the center.
  • f(r)>0f(r) > 0→\rightarrow pushing away from the center.
  • r^\hat{r}== radial unit vector.
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