Answer all clearly and correctly for five stars and best answer. A long, straigh

Question

Answer all clearly and correctly for five stars and best answer.

A long, straight conducting wire of radius R has a nonuniform current density J = Jo r3/R4 where Jo is a constant. The wire carries a total current I. Surrounding the straight conducting wire is a hollow cylinder of inner radius Ra and outer radius Rb with a uniform current density. The hollow cylinder carries a total current I traveling in the same direction as the conducting wire. Find the expression for Jo in terms of I and R. Find the magnetic field (a) inside the wire, (b) in the space between the wire and the hollow cylinder (c) between the hollow cylinder, and (d) outside the hollow cylinder. Draw the B vs graph for the entire regions solved above. Draw the sketch, provide reasoning and check your answer.

Explanation / Answer

I = int{J da}

I = int{(J0 r^3/R^4) (2 pi r dr)} ; from r=0 to r=R

I = (2 pi J0/R^4) int{r^4 dr} ; from r=0 to r=R

I = (2 pi J0/R^4) (r^5/5) ; from r=0 to r=R

I = (2 pi J0/R^4) (R^5/5)

==> I = 2 pi J0 R/5

==> J0 = 5 I/(2 pi R)

----------------------------------------------------------------------------------

a)

i = int{(J0 r'^3/R^4) (2 pi r' dr')} ; from r=0 to r'=r

I = (2 pi J0/R^4) int{r'^4 dr'} ; from r=0 to r'=r

I = (2 pi J0/R^4) (r'^5/5) ; from r=0 to r'=r

I = (2 pi J0/R^4) (r^5/5)

B = u0 i/(2 pi r) = u0 ((2 pi J0/R^4) (r^5/5))/(2 pi r) = u0 ((J0/R^4) (r^4/5)) = (u0 J0/R^4) (r^4/5)

==> B = u0 I r^4/(2 pi R^5)

b)

B = u0 I/(2 pi r)

c)

i2 = I ((r)^2 - (Ra)^2))/((Rb)^2 - (Ra)^2))

i_total = I + i2 = I [1 + ((r)^2 - (Ra)^2))/((Rb)^2 - (Ra)^2))]

B = u0 (i_tot)/(2 pi r) = u0 I [1 + ((r)^2 - (Ra)^2))/((Rb)^2 - (Ra)^2))]/(2 pi r)

d)

B = u0 (2 I)/(2 pi r) = u0 I/(pi r)

-----------------------------------------------------------------------------------

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.