Biot-Savart Law (infinite filament)

[kingmaker]

biot savart law finite wire

An infinite current filament carries a current of 3A and lies along the x-axis. Using Biot-Savart Law, find the magnectic field intensity in cartesian coordinates at a point P(-1,3,2).

dH = I vec{dl} x hat{R} / 4piR^2

let substitude hat{R} with vec{R} / R

then dH = I vec{dl} x vec{R} / 4piR^3

vec{R} = hat{x}(-1-x) + hat{y}3 + hat{z}2 and vec{dl} = hat{x}dx

vec{dl} x vec{R} =
|hat{x} hat{y} hat{z}|
| 1 0 0 |
| (-1-x) 3 2 |

=[hat{y}-2 + hat{z}3] dx

dH = (I)(dx)(hat{y}-2 + hat{z}3) / 4piR^3

magnitude of R = sqrt [(-1-x)^2 + 3^2 + 2^2]

dH = (I)(dx)(hat{y}-2 + hat{z}3) / 4pi[(-1-x)^2 + 3^2 + 2^2]^3/2

how do I solve for the next step? thanks!

 

[subharpe]

biot-savart infinite wire

Because it is an infinite wire use the BS formulation of finite straight wire to the case of infinite wire. Here because the wire is along the x-axis, the x-ordinate (1) is of no importance, as any x-ordinate is identical w.r.t. infinite wire.

The distance of the field point from the wire is R = (2^2 + 3^2)^0.5 = √13.

The formula to be used (which you can verify) is

H = (i/4ΠR)(cos α + cos β) where α, β are the angles the field point makes at the end points of the wire, which are obviously Π/2 both. Therefore the Hfield is

H = i/2ΠR.

You can use Ampere's law directly in this case too, which also gives the same result.

 

[xulfee]

probebly u've to integrate nxt to tht