Q19P - Introduction to Electrodynamics

Question

Draw a circle in the xy plane. At a few representative points draw the vector v tangent to the circle, pointing in the clockwise direction. By comparing adjacent vectors, determine the sign of $\mathbf{\partial }{\mathit{v}}_{\mathbf{x}}\mathbf{ }\mathit{l}\mathbf{\partial }\mathit{y}$and $\mathbf{\partial }{\mathit{v}}_{\mathbf{y}}\mathbf{ }\mathit{l}\mathbf{ }\mathbf{\partial }\mathit{x}$According to Eq. 1.41, then, what is the direction of  $\mathbf{\nabla }\mathbf{\times }\mathit{v}$? Explain how this example illustrates the geometrical interpretation of the curl.

Step-by-Step Solution

Answer

The tangential vector for circular motion in the clock wise direction is drawn as follows:

                                     

                                                 

The direction of curl of v is in -z direction.

Step 1: Describe the given information.

It is given that a circle is in the xy plane. At a few representative points, the vector v is tangent to the circle, pointing in the clockwise direction, then the direction of curl of v is to be evaluated.

Step 2: Define the velocity vector.

The velocity vector is obtained as the gradient of the scalar distance function$\mathit{d}\mathbf{ }\mathbf{(}\mathit{x}\mathbf{,}\mathit{y}\mathbf{,}\mathit{z}\mathbf{)}$. For circular motion the curl of velocity is 0.

Step 3: Compute the curl of vector v.

Sketch the tangential vector for circular motion in the clock wise directions follows:

                                 

At four points shown, 1, 2, 3, and 4, the direction of the movement of the velocity vector is shown.

On moving from point 1 to 2, there is an increase in y direction and decrease in x direction. So, the velocity in x and y direction must be increased that is $v_{x_{,}} v_y$must be increased.

Similarly, on moving from point 3 to 4, there is an decrease in y direction and increase in x direction So, the velocity in x and y direction must be decreased that is $v_{x_{,}} v_y$ must be decreased. Hence, we can write

$\frac{\partial v_y}{\partial x}<0$, $\frac{\partial v_x}{\partial y}>0$.

Step 4: Compute the curl of v.

Compute curl of vector$v$.

$$\begin{gathered} \nabla \times v=\nabla \times (v_{x}i+v_y j) \\ =\left|\begin{array}{ccc}i& j& k\\ \frac{\partial }{\partial x}& \frac{\partial }{\partial y}& \frac{\partial }{\partial z}\\ v_x& v_y& 0\end{array}\right| \\ =i (0-0)-j(0-0)+k\left(\frac{\partial v_y}{\partial x}-\frac{\partial v_x}{\partial z}\right) \\ =k\left(\frac{\partial v_y}{\partial x}-\frac{\partial v_x}{\partial y}\right) \end{gathered}$$

As the value of $\frac{\partial v_y}{\partial x}$is negative and value of $\frac{\partial v_x}{\partial y}$ is positive, the curl of  is overall negative and points in $-z$ direction.

Step 5: Compute the direction curl of v using right hand rule.

According to the right hand rule, the direction of closed four finger of right fist, represents the direction of tangential velocity vector, which is in clock wise direction, whereas the direction curl is represented by the thumb , which is into the right circle

Thus, the direction of curl of v is in -z direction.