# Find the magnetic field at the point

• Stendhal
In summary, the wire produces a magnetic field of 8.4*10^{-15} T at the points x=2 meters, y=2 meters, and z=0.
Stendhal

## Homework Statement

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.

## Homework 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)

## 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!

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.

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?

Stendhal said:
the function is odd, not even.
No, it is even.
Stendhal said:
would it be possible to solve this using Ampere's Law to find the magnitude?
Just drop the s2 in the denominator, making it constant over the range. Assuming everything else is right.

## What is magnetic field and why is it important?

Magnetic field is a region in space where a magnetic force is exerted on a charged particle. It is important because it helps us understand the behavior of magnets and their interactions with other objects.

## How is the magnetic field at a point calculated?

The magnetic field at a point is calculated using the formula B = μ0 * (I / 2πr), where μ0 is the permeability of free space, I is the current, and r is the distance from the point to the current-carrying wire.

## What factors affect the strength of the magnetic field at a point?

The strength of the magnetic field at a point can be affected by the current flowing through a wire, the distance from the wire, the material of the wire, and any external magnetic fields present.

## What is the unit of measurement for magnetic field?

The unit of measurement for magnetic field is Tesla (T) in the SI system. In some cases, Gauss (G) is also used, where 1 T = 10,000 G.

## What are some real-life applications of finding the magnetic field at a point?

Some real-life applications include designing and constructing electromagnets, motors, generators, and other electrical devices. It is also important in fields such as geology, where it helps in mapping the Earth's magnetic field.

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