Problem given was:
- One end of a long, straight, current-carrying metal wire is electrically connected to a very large, thin, homogeneous metal plate, whose plane is perpendicular to the wire, and whose distant perimeter is earthed. One of the terminals of the battery supplying the current is also earthed.
Ignoring the Earth's field, find the magnitude and direction of the magnetic field at a point $P$ that is a distance $r$ from the wire and at a height $h$ above the metal plate, when a steady current $I$ flows in the wire (see figure).
Describe the magnetic field above and below the plate.
Solution given says this:
Let the unit vector $\boldsymbol{e}$ represent direction of current in the wire, and $\boldsymbol{R}$ be the position vector of $P$ relative to the junction of the wire and plate (see Figure). The magnetic field vector must be determined by these two vectors (and the given current magnitude $I$ ) as a single-valued function $\boldsymbol{B}(\boldsymbol{e}, \boldsymbol{R})$, i.e. the magnetic field is a vector function of two vector variables.
To find the direction of the magnetic field, the 'right-hand (corkscrew) rule' must play a central role, and this fact greatly constrains the form of the function $\boldsymbol{B}(\boldsymbol{e}, \boldsymbol{R})$. Using just these two vectors, the only formulation for a magnetic field vector that is in accord with the right-hand rule is their vector product. For this reason, the magnetic field vector must be proportional to $\boldsymbol{e} \times \boldsymbol{R}$, and the proportionality factor must be a scalar function that does not depend upon the (arbitrary) sign convention in the right-hand rule: $\boldsymbol{B}(\boldsymbol{e}, \boldsymbol{R})=(\boldsymbol{e} \times \boldsymbol{R}) f\left(\boldsymbol{e} \cdot \boldsymbol{R}, R^2, I\right)$ after this it was told that scalar function for some fixed values of $r$ and $h$ has constant value so thus it showed its independent of azimuthal position . Now from this magnetic field direction can be easily deduced . My query is related to the fact they argued that
The form of B direction should be of $\boldsymbol{e} \times \boldsymbol{R}$ but why not it can be of form like $\boldsymbol{e}$ $\times$ ($\boldsymbol{e} \times \boldsymbol{R}$) or as in more such cross products using $\boldsymbol{e}$ and $\boldsymbol{R}$? Why only that form ?
- 2)What they meant by this : "proportionality factor must be a scalar function that does not depend upon the (arbitrary) sign convention in the right-hand rule" ? Why we want that and also isnt the right hand rule gives a single direction ? Why arbitrary ?