# Find the magnetic field at the point

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1. Mar 27, 2017

### Stendhal

1. The problem statement, all variables and given/known data
A long, straight wire lies along the z−axis and carries a 4.20 −A current in the +z−direction. Find the magnetic field (magnitude and direction) produced at the following points by a 0.400 −mm segment of the wire centered at the origin.

2. Relevant equations

$\vec B$ = 2 $\frac {μ_0} {4π}$ $\int_0^.0004$ $\frac {I*d\vec s \times \hat r} {r^2}$

x = 2 meters, y = 2meters, z=0

$d\vec s \times \hat r$ = ds*sin(Θ)

3. The attempt at a solution
I took the original equation, which is from the Biot-Savart Law, and simplified the cross product using the algebraic definition above, such that the problem simplifies into:

$\vec B$ = 2 $\frac {μ_0 I} {4π}$ $\int_0^.0004$ $\frac {ds*sin(Θ)} {r^2}$

Where the angle is that between the z-axis and the hypotenuse, such that sin(Θ) can be described as:
sin(Θ) = $\frac {\sqrt (x^2+y^2)} {r}$

which gives

$\vec B$ = 2 $\frac {μ_0 I \sqrt(8)} {4π}$ $\int_0^.0004$ $\frac {ds} {\sqrt(s^2 +2^2 +2^2)^3}$

Which, when I solve gives me the answer for the magnitude and direction of the magnetic field of
$8.4*10^{-15} T$ which doesn't seem right at all, since previous answers to some other parts of this problem were in around ^-11. Any help is greatly appreciated!

2. Mar 28, 2017

### haruspex

It says centered at the origin, so I think your integration range is wrong.

The wire length is so small compared to √8 m that I don't think you need to integrate at all. Just treat the whole segment as being at the origin.

3. Mar 28, 2017

### Stendhal

Since it it centered at the origin, the integration would be $\int_-.0002^.0002$ which I believed could be changed to 2 $\int_0^.0004$ Though that is wrong because the function is odd, not even.

Also, since the wire length is so small compared to $\sqrt 8$, would it be possible to solve this using Ampere's Law to find the magnitude?

4. Mar 28, 2017

### haruspex

No, it is even.
Just drop the s2 in the denominator, making it constant over the range. Assuming everything else is right.