The current density varies with radial distance r as J= ar2, in a cylindrical wire of radius R. The current passing through the wire between radial distance R/3 and R/2 is,

Question

The current density varies with radial distance r as J= ar2, in a cylindrical wire of radius R. The current passing through the wire between radial distance R/3 and R/2 is, (A) (65 π a R4/2592) (B) (25 π a R4/72) (C) (65 π a R3/2938) (D) (81 π a2 R4/144)

Tardigrade Admin · Accepted Answer

Current density J = ar2 Current I= ∫ J dA I=∫ limitsR/2R/3 a r2(2π rdr), [where dA = 2π rdr] =2π a[(r4/4)]R/2R/3=(65 π a R4/2592)

Q. The current density varies with radial distance $r$ as $J = a r^{2}$ , in a cylindrical wire of radius $R$ . The current passing through the wire between radial distance $R /3$ and $R /2$ is,

$\frac{65 π a R ^{4}}{2592}$

$\frac{25 π a R ^{4}}{72}$

$\frac{65 π a R ^{3}}{2938}$

$\frac{81 π a ^{2} R ^{4}}{144}$

Current density $J = a r^{2}$
Current $I = \int J d A$
$I = R /3 \int R /2 a r^{2} (2 π r d r),$ [where $d A = 2 π r d r$ ]
$= 2 πa [\frac{r ^{4}}{4}]_{R /3}^{R /2} = \frac{65 π a R ^{4}}{2592}$

Q. The current density varies with radial distance r as J=ar2, in a cylindrical wire of radius R. The current passing through the wire between radial distance R/3 and R/2 is,

259265πaR4​

7225πaR4​

293865πaR3​

14481πa2R4​