AP EAMCET · PHYSICS · Current Electricity

Current density in a cylindrical wire of radius $R$ varies with radial distance as $β {(r + r_{0})}^{2}$ . The current through the section of the wire shown in the figure is.

A $π β [\frac{R^{4}}{12} + \frac{r_{\circ}^{2} R^{2}}{6} + \frac{2 r_{0} R^{3}}{9}]$
B $π β [\frac{R^{4}}{6} + \frac{r_{0}^{2} R^{2}}{12} + \frac{r_{0} R^{3}}{9}]$
C $π β [\frac{R^{4}}{12} + \frac{r_{0}^{2} R^{2}}{12} + \frac{r_{0} R^{3}}{9}]$
D $π β [\frac{R^{4}}{8} + \frac{r_{0}^{2} R^{2}}{6} + \frac{r_{0} R^{3}}{12}]$

Verified Solution

Answer & Solution

Correct Answer

(A) $π β [\frac{R^{4}}{12} + \frac{r_{\circ}^{2} R^{2}}{6} + \frac{2 r_{0} R^{3}}{9}]$

Step-by-step Solution

Detailed explanation

Current density is current per unit area. If we take a circular element of radius r and thickness dr, the current through it will be, di=2πrdrβr+r02⇒di=2πrdrβr2+2r0r+r02⇒di=2πβr3dr+2r0r2dr+r02rdr Integrating the above equation from 0 to…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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Current density in a cylindrical wire of radius R varies with radial distance as βr+r02. The current through the section of the wire shown in the figure is.

Step-by-step Solution

Current density in a cylindrical wire of radius $R$ varies with radial distance as $β {(r + r_{0})}^{2}$ . The current through the section of the wire shown in the figure is.