Finding the Covariant Components of a Magnetic Field

In non-Cartesian coordinate systems, the basis vectors are not of unit length, causing the components to behave differently. The process uses the metric tensor as a conversion tool, which accounts for the varying scale of the coordinate system. The result shows that while the initial contravariant component $\left(B^\phi\right)$ depends on the radial distance $(\rho)$ from the wire, the final covariant component ( $B_\phi$ ) is constant, directly representing the physical field strength after accounting for the basis vector's length.

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✍️Mathematical Proof

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Distinction Between Components: In non-Cartesian coordinate systems like cylindrical coordinates, the basis vectors are not always unit vectors and can change magnitude with position. The contravariant components ( $B^\rho, B^\phi, B^z$ ) are coefficients used to represent the vector in terms of these basis vectors. The covariant components ( $B_\rho, B_\phi, B_z$ ) are projections of the vector onto the basis vectors, representing the physical field strength.

Role of the Metric Tensor: The metric tensor, $g_{i j}$, acts as a conversion tool. It is the core object that contains information about the geometry of the coordinate system. Multiplying the contravariant components by the metric tensor correctly scales them to yield the covariant components, reflecting the true physical behavior of the field.

Physical Significance: The result of the calculation, $B_\phi=\frac{\mu_0 I}{2 \pi}$, shows that the covariant component is independent of $\rho$. This is counterintuitive at first, but it makes sense when you consider that the basis vector $e_\phi$ has a length of $\rho$. This means that the product $B^\phi \cdot e_\phi$ (contravariant component times basis vector) gives the physical field magnitude, which correctly decreases with distance from the wire as $\frac{\mu_0 I}{2 \pi \rho}$. The covariant component, $B_\phi$, effectively absorbs this $\rho$ dependence into its definition, making it constant. This demonstrates why covariant components often have a more direct physical interpretation in these curved coordinate systems.

✍️Mathematical Proof

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🧄Proof and Derivation-1

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