Question

The magnetic flux through each of five faces of a die (singular of “dice”) is given by ${\mathit{\phi }}_{\mathbf{B}}\mathbf{=}\mathbf{\pm }\mathit{N}\mathit{W}\mathit{b}$, where N(= 1 to 5) is the number of spots on the face. The flux is positive (outward) for Neven and negative (inward) for Nodd. What is the flux through the sixth face of the die?

Short answer

Magnetic flux through the sixth face is 3 Wb.

Step-by-step solution

Given

1. Magnetic flux through each of five faces of a die is${\phi }_b=\pm NWb$.
2. For even N, flux is positive (outward).
3. For oddN, the flux is negative (inward).

Determining the concept

Using Gauss law, write an expression for net magnetic flux through the die. Inserting the magnetic flux through five faces in it gives magnetic flux through the sixth face of the die.

Determining the magnetic flux through the sixth face

The dice with 6 sides forms a closed Gaussian surface. Using Gauss law for magnetism, the net magnetic flux through any closed surface is zero.

${\phi }_b=\oint B.dA=0............................\left(1\right)$

For 6 faces of the die,

$$\begin{gathered} {\phi }_b=\sum _{N=1}^6{\phi }_{bN}=0 \\ {\phi }_{b1}+{\phi }_{b2}+{\phi }_{b3}+{\phi }_{b4}+{\phi }_{b5}+{\phi }_{b6}=0. \end{gathered}$$

It is given that for odd N, flux is negative and for even N, flux is positive.

$\begin{array}{rcl}-1Wb+2Wb-3Wb+4Wb-5Wb+{\phi }_{b6}& =& 0\\ -9Wb+6Wb+{\phi }_{b6}& =& 0\\ {\phi }_{b6}& =& -3Wb\end{array}$

Hence, the magnetic flux through the sixth face is - 3 Wb.