# Problem with vectors and unit vectors

#### TW Cantor

1. Homework Statement

An electric charge e which moves through a material of permeability μ with a velocity v, produces a magnetic field β at a point r given by:

β= ((μ*e)/(4*∏))*((v*ru)/(|r|)2)

where ru is a unit vector in the direction of r.

Find β if v=4i+3j+3k and r=4i+7j+5k.
Take μ=e=93 in appropriate units.

2. Homework Equations

equation for a unit vector: (a*b)=(a2*b3-a3*b2)i-(a1*b3-a3*b1)j-(a1*b2-a2*b1)k

3. The Attempt at a Solution

so i know μ .and e are both = 93 so i know that:
(μ*e)/(4*∏) is going to be a constant. i worked it out to be 688.266

i worked out |r|2:
42+72+52=90

then i tried to use the equation for a unit vector to calculate ru. this gave me:
ru = -48.902i + 0.060j - 65.143k

i then put all these values back into the original equation and it came out with:
β = -373.973i + 0.459j - 498.174k
β = -4.836i - 6.448j + 12.897k

can anyone help me find what i have done wrong?

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#### cupid.callin

Isnt μ/4∏ = 10^-7???

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#### TW Cantor

By the * i mean multiply.
so im told μ and e both = 93 then μ*e = 93*93 = 8649
then i divided this by 4∏:
8649/4∏ = 688.265 with my calculator in radians. so i presumed that that section of the problem would always be constant.

what units should i be using for μ and e?

#### cupid.callin

well you can use SI for all
In SI

μ/4∏ = 10^-7
e = 1.6 x 10^-19

but if its given then its a completely different matter

but my answer is coming something different ...

#### TW Cantor

so have i gone wrong working out (μ*e)/4∏ then?

i work out v*r to be -6i - 8j - 16k

to calculate ru, what would be the correct method?
would it be r/|r| ?

#### cupid.callin

OH!!! I'm sorry its a unit vector!!!! i didnt look at (u)

ru = r/mag(r)

SO v*r = -6i - 8j - 16k

v*ru = (-6i - 8j - 16k)/sqt(90)

or you can write: B = 288.266/(90 x sqrt(90)) * (-6i - 8j - 16k)

so you get ...

β = -4.836i - 6.448j + 12.897k

#### TW Cantor

haha thats fine :-)
i understand what you have done but why is the k component of the vector positive?

#### cupid.callin

i actually copy pasted from your post and sisnt looked so closely but k component comes to be positive. Check your calculations.
and you can also use matrix method to find cross product
it has less chances of making sign mistake

#### TW Cantor

so i have got the k component of (v*ru) to be -1.6865

if i divide this by 90 i get -0.0187

then multiply by 688.266 and i get -12.897?

did you get -1.6865 to be the k component? or is it positive? im not yet familiar with the matrix method

#### cupid.callin

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