Question

The magnetic flux through each of five faces of a die (singular of ''dice'') is given by Φ_B = ±N Wb, where N (= 1 to 5) is the number of spots on the face. The flux is positive (outward) for N even and negative (inward) for N odd. What is the flux through the sixth face of the die?

Answer 1

Let flux through sixth face is Wb

Total magnetic flux = 0

3 Wb ...... Answer

Answer 2

Calculating the Flux Through the Sixth Face of the Die

Given:

• A die has 6 faces, labeled with $N=1$ to $6$ spots.

• Magnetic flux through five faces is given by:

• ${\mathrm{\Phi }}_B=+N\text{Wb}$ (outward) if $N$ is even.

• ${\mathrm{\Phi }}_B=-N\text{Wb}$ (inward) if $N$ is odd.

• The flux through the sixth face (unknown) is denoted as ${\mathrm{\Phi }}_6$.

Key Principle: Gauss's Law for Magnetism

$$\oint \mathbf{B}\cdot d\mathbf{A}=0$$

This means the total magnetic flux through a closed surface (like the die) must be zero.

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Step 1: Calculate Flux Through the Five Known Faces

Face ($N$)	Flux (${\mathrm{\Phi }}_B$)	Reason
1	$-1\text{Wb}$	$N=1$ is odd.
2	$+2\text{Wb}$	$N=2$ is even.
3	$-3\text{Wb}$	$N=3$ is odd.
4	$+4\text{Wb}$	$N=4$ is even.
5	$-5\text{Wb}$	$N=5$ is odd.
Total (5 faces)	$-3\text{Wb}$	Sum: $-1+2-3+4-5=-3$.

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Step 2: Solve for the Sixth Face ($N=6$)

Using Gauss's Law:

$${\mathrm{\Phi }}_{\text{total}}={\mathrm{\Phi }}_{\text{5 faces}}+{\mathrm{\Phi }}_6=0$$$$-3\text{Wb}+{\mathrm{\Phi }}_6=0⟹{\mathrm{\Phi }}_6=+3\text{Wb}$$

Verification:

• The sixth face has $N=6$ (even), so by the given rule:

  $${\mathrm{\Phi }}_6=+6\text{Wb}\text{(if it followed the pattern)}$$

• However, Gauss's Law overrides this pattern, requiring the total flux to be zero. Thus:

  $${\mathrm{\Phi }}_6=+3\text{Wb}$$

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Answer:

The flux through the sixth face is $+3\text{Wb}$ (outward).