Defining the current vector in the biot savart law?

[arronslacey]

I am trying to use the biot savart law to calculate the magnetic field of a given object. I have got to the stage where I have calculated I*dl and R/R^2 separately (doing this in matlab. The problem is where I come to the cross product. If I have a uniform current, the values of the current vector would be zero where there is no cable. i.e. if I have a current loop of uniform current = 1, anywhere outside or inside the current loop, the value of I*dl = 0 right? So if this is correct, when I take the cross product of I*dl and R/R^2, I will be crossing a vector of value 0, with the R/R^2 in places outisde of the loop, which leads to a value of 0. Although, the magnetic field due to the current is only 0 is the distance goes to infinity, so I cannot have a space in the vicinity of the wire with magnetic field = 0. What am I not understanding here?

 

[tiny-tim]

hi arronslacey!

the biot-savart law is B = (µ_o/4π) ∫ (I dl x r^)/r²

it gives the magnetic field induced by a current I flowing along a wire with line lement dl

you only use it along the wire! ​

 

[arronslacey]

HI Tim, thanks for you reply. I see that you only use the current on the actual wire. I'll try to explain a bit further. I am doing this in matlab, so each variable in the equation is in the form of a matrix. I have a picture of a circle which I am trying to super impose a magnetic field on. So the variables might look like:

I = 0 0 0 0 0 0
0 0 1 1 0 0
0 1 0 0 1 0
0 0 1 1 0 0
0 0 0 0 0 0

dL = 0 0 0 0 0 0
0 0 -0.05 0.05 0 0
0 -0.05 0 0 0.05 0
0 0 -0.05 0.05 0 0
0 0 0 0 0 0

where I need to cross dL with R = Rs/Rxs.^2. Doing a cross product will take element (1,1) of dL and cross it with element R(1,1), which would give me 0. This should not be the case! so either my logic is wrong here, or I am using the wrong variables in the cross product.

 

[tiny-tim]

(isn't I just a number? )

r is the position vector from the element dl to the fixed point that you're measuring B at

 

[Nickie]

The current vector in the Biot-Savart law is a representation of the direction and magnitude of the current flow. It is a vector quantity, meaning it has both magnitude and direction. In the context of calculating the magnetic field using the Biot-Savart law, the current vector represents the direction of the current flow at a specific point in space.

In your example, it is correct that the value of I*dl would be zero outside of the current loop. However, this does not mean that the current vector is equal to zero. The current vector will still have a direction, even if the magnitude is zero. This is because the direction of the current flow is determined by the direction of the current itself, not the magnitude.

When taking the cross product of I*dl and R/R^2, you are not crossing a vector of value 0 with R/R^2. The value of I*dl may be zero, but the direction of the current vector is still present and is important in determining the direction of the resulting magnetic field.

Additionally, it is important to note that the magnetic field due to a current is not only zero at infinity, but also at points on the axis of symmetry of the current loop. This means that there will be points near the wire where the magnetic field is zero, but this does not mean that the entire space around the wire has zero magnetic field.

In summary, the current vector in the Biot-Savart law represents the direction of the current flow, and even if the magnitude is zero, the direction is still present and important in determining the resulting magnetic field. The magnetic field due to a current is not only zero at infinity, but also at specific points near the wire, but this does not mean that the entire space around the wire has zero magnetic field.