Concept:For a thick, uniform current-carrying cylindrical wire, the magnetic field behavior differs inside the wire compared to outside the wire. This is determined using Ampere's Law.
Formula:$$ \oint B \cdot dl = \mu_0 I_{\text{enc}} $$
Solution:- Inside the wire (at a radius \( r < R \), where \( R \) is wire radius), the current enclosed by an Amperian loop depends on the area: \( I_{\text{enc}} = I \frac{r^2}{R^2} \).
- Apply Ampere's law: \( B(2\pi r) = \mu_0 I \frac{r^2}{R^2} \).
- Solve for \( B \): \( B = \left(\frac{\mu_0 I}{2\pi R^2}\right) r \).
- Because all terms in the parenthesis are constants, the magnetic field strictly increases linearly (directly) with r from the center to the surface.
Why other options are incorrect:Option A is true for the field
outside the wire. Options B and D represent completely incorrect mathematical relationships.
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