Problem with vectors and unit vectors

In summary, the conversation discusses the calculation of the magnetic field produced by an electric charge moving through a material with a given velocity and permeability. The equation for the magnetic field is given, and the necessary values for μ and e are provided. The conversation also includes a discussion on finding the unit vector in the direction of r and the use of the matrix method to find the cross product. The final answer for the magnetic field is given and compared to the calculated answer."
  • #1
TW Cantor
54
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.



Homework Equations



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


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
when the actual answer is:
β = -4.836i - 6.448j + 12.897k

can anyone help me find what i have done wrong?
 
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  • #2
Isnt μ/4∏ = 10^-7?
 
Last edited:
  • #3
By the * i mean multiply.
so I am 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?
 
  • #4
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 ...

whats your product of v*r
 
  • #5
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| ?
 
  • #6
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
 
  • #7
haha that's fine :-)
i understand what you have done but why is the k component of the vector positive?
 
  • #8
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
 
  • #9
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? I am not yet familiar with the matrix method
 
  • #10
Last edited by a moderator:

What are vectors and unit vectors?

Vectors are mathematical quantities that have both magnitude and direction. They are often represented as arrows in a coordinate system. Unit vectors are vectors with a magnitude of 1 and are commonly used to represent directions or orientations.

What is the difference between a vector and a unit vector?

The main difference between a vector and a unit vector is their magnitude. Vectors can have any magnitude and direction, while unit vectors have a magnitude of 1 and are used to represent specific directions or orientations.

How do you calculate the magnitude of a vector?

The magnitude of a vector can be calculated using the Pythagorean theorem, where the magnitude is equal to the square root of the sum of the squares of the vector's components. In other words, the magnitude is the length of the vector.

What does it mean for two vectors to be parallel?

Two vectors are parallel if they have the same direction or are in the same line, even if they have different magnitudes. This means that the vectors are either pointing in the same direction or in exact opposite directions.

How can unit vectors be used to represent directions?

Unit vectors can be used to represent directions by assigning a specific unit vector to a specific direction. For example, the unit vector i represents the x-axis, j represents the y-axis, and k represents the z-axis. By combining these unit vectors, any direction in 3D space can be represented.

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