Past exam of the mathematics course of the University of Cambridge / 2020 / ib / Paper 2 / 15D / b

Table of contents
- Solution b

Solution

The accumulated point charge

Q (t)

produces

E (r, t) = \frac{Q ( t )}{4 π ϵ _{0} r ^{2}} e_{r} .

Since

\dot{Q} = I

, the displacement current density is

ϵ_{0} \frac{\partial E}{\partial t} = \frac{I}{4 π r ^{2}} e_{r} .

Apply the integral Ampère-Maxwell equation to the boundary of a spherical cap of radius

r

and polar angle

θ < π /2

. By axial symmetry

B = B_{ϕ} e_{ϕ}

is constant along the boundary, whose circumference is

2 π r sin θ

. The displacement-current flux through the cap is

\int_{cap} \frac{I}{4 π r ^{2}} d A = \frac{I}{4 π r ^{2}} 2 π r^{2} (1 - cos θ) = \frac{I}{2} (1 - cos θ) .

Thus

2 π r sin θ B_{ϕ} = \frac{μ _{0} I}{2} (1 - cos θ) .

Using

(1 - cos θ) / sin θ = tan (θ /2)

gives

B (r, θ) = \frac{μ _{0} I}{4 π r} tan (\frac{θ}{2}) e_{ϕ} .

 Ancestors (10)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version