4 Engineering Example 1

4.1 Current associated with a magnetic field

Introduction

In a magnetic field $\underset{̲}{B}$ , an associated current is given by:

$\underset{̲}{I}=\frac{1}{{\mu }_0}\left(\underset{̲}{\nabla }\times \underset{̲}{B}\right)$

Problem in words

Given the magnetic field $\underset{̲}{B}=B_{0}x\underset{̲}{k}$ find the associated current $\underset{̲}{I}$ .

Figure 19:

Mathematical statement of problem

We need to evaluate the curl of $\underset{̲}{B}$ . Mathematical analysis

and so $\underset{̲}{I}=-\frac{B_0}{{\mu }_0}\underset{̲}{j}$ .

Interpretation

The current is perpendicular to the field and to the direction of variation of the field.

Task!

Find the curl of the following two-dimensional vector field

1. in general terms and
2. at the point $\left(1,2\right)$ .

   ${\underset{̲}{F}}_2=y^2\underset{̲}{i}+xy\underset{̲}{j}$

Answer

Exercises

1. Find the curl of each of the following two-dimensional vector fields. Give each in general terms and also at the point $\left(1,2\right)$ .
   1. ${\underset{̲}{F}}_1=2x\underset{̲}{i}+2y\underset{̲}{j}$
   2. ${\underset{̲}{F}}_3=x^2{y}^3\underset{̲}{i}-x^3{y}^2\underset{̲}{j}$
2. Find the curl of each of the following three-dimensional vector fields. Give each in general terms and also at the point $\left(2,1,3\right)$ .
   1. ${\underset{̲}{F}}_1=y^2{z}^3\underset{̲}{i}+2xy{z}^3\underset{̲}{j}+3x{y}^2{z}^2\underset{̲}{k}$
   2. ${\underset{̲}{F}}_2=\left(xy+z^2\right)\underset{̲}{i}+x^2\underset{̲}{j}+\left(xz-2\right)\underset{̲}{k}$
3. The surface water velocity on a straight uniform river 20 metres wide is modelled by the vector $\underset{̲}{v}=\frac{1}{50}x\left(20-x\right)\underset{̲}{j}$ where $x$ is the distance from the west bank (see diagram).

   

   1. Find the velocity $\underset{̲}{v}$ at each bank and at midstream.
   2. Find $\underset{̲}{\nabla }\times \underset{̲}{v}$ at each bank and at midstream.
4. The velocity field on the surface of an emptying bathroom sink can be modelled by two functions, the first describing the swirling vortex of radius $a$ near the plughole and the second describing the more gently rotating fluid outside the vortex region. These functions are

   $\underset{̲}{u}\left(x,y\right)=w\left(-y\underset{̲}{i}+x\underset{̲}{j}\right)$ ,      $\left(\sqrt{x^2+y^2}\le a\right)$

   $\underset{̲}{v}\left(x,y\right)=\frac{w{a}^2\left(-y\underset{̲}{i}+x\underset{̲}{j}\right)}{x^2+y^2}$      $\left(\sqrt{x^2+y^2}\ge a\right)$

   Find

   1. curl $\underset{̲}{u}$ and
   2. curl $\underset{̲}{v}$ .

Answer