AP EAMCET20224 Jul 2022Morning ShiftPhysicsCurrent ElectricityActual
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.
Options
- Aπ β R 4 12 + r ∘ 2 R 2 6 + 2 r 0 R 3 9
- Bπ β R 4 6 + r 0 2 R 2 12 + r 0 R 3 9
- Cπ β R 4 12 + r 0 2 R 2 12 + r 0 R 3 9
- Dπ β R 4 8 + r 0 2 R 2 6 + r 0 R 3 12
Correct answer
A. π β R 4 12 + r ∘ 2 R 2 6 + 2 r 0 R 3 9
Step-by-step solution
Current density is current per unit area. If we take a circular element of radius r and thickness d r , the current through it will be, d i = 2 π r d r β r + r 0 2 ⇒ d i = 2 π r d r β r 2 + 2 r 0 r + r 0 2 ⇒ d i = 2 π β r 3 d r + 2 r 0 r 2 d r + r 0 2 r d r Integrating the above equation from 0 to R we will get current through the complete wire. ∫ 0 I d i = ∫ 0 R 2 π β r 3 d r + 2 r 0 r 2 d r + r 0 2 r d r ⇒ I = 2 π β R 4 4 + 2 r 0 R 3