# Angle between current and magnetic field

ok, F = BIL sin(theta)

B= mag field direction,
I = current,
L = length of wire,
F = induced force by magnetic field on the current carrying wire in the field

so why does the induced force tend to increase at a much lesser amount when the angles become bigger, i.e around 70 - 90degrees.

i plotted the graph of Force against angle and found that at low angles, it is approximately linear. but at high angles, it sort of curve downwards, forming a concave maximum point graph.

is this due to the inherent sin(theta) characteristic , or is it because of the fact that nature is as it is, and hence we use sin(theta) and not (theta) for the formula?

so if its nature, then is there any explanation as to why high angles do not behave linearly with the induced force?

thanks a lot!

tiny-tim
Homework Helper
Hi quietrain! Draw a vertical square perpendicular to a horizontal field.

Now tilt the square at an angle whose sine is 1/n.

Make n copies of that tilted square (half of them titled the other way), to make a "concertina" …

that concertina will reach the same height as the original square, and so will intersect exactly the same amount of the field,

so the amount of field through each individual square will be 1/n times the original amount,

ie sine times the original amount. The force on a length of wire L carrying a current I in a magnetic field B is given by the Lorentz force equation

F = L(I x B)

where I x B is the vector cross product of I and B, and has a magnitude I·B sin(θ), where θ is the angle between I and B, and the force is perpendicular to the plane including both I and B. For an elegant physics lab demonstration of this force, see

https://www.physicsforums.com/attachment.php?attachmentid=22954&d=1263085690

This photograph shows a 300-volt electron beam in a vacuum tube being deflected by a magnetic field in a Helmholtz coil.

Bob S

tiny-tim
Homework Helper
… the Lorentz force equation …

Hi Bob! Isn't that Faraday's equation? Faraday's Law of induction in integral form is

E·dl = -(d/dt)∫B·n dA

Its key feature is the time derivative, leading to a dB/dt in most cases (as in transformers).

Bob S

Hi quietrain! Draw a vertical square perpendicular to a horizontal field.

Now tilt the square at an angle whose sine is 1/n.

Make n copies of that tilted square (half of them titled the other way), to make a "concertina" …

that concertina will reach the same height as the original square, and so will intersect exactly the same amount of the field,

so the amount of field through each individual square will be 1/n times the original amount,

ie sine times the original amount. ok so this is how sine function comes about right?

but, so is there any explanation as to why at high angles, the FORCE VS ANGLE graph is not linear, for a wire in a magnetic field? but rather, a curve? whereas at low angles, its approximately linear?

is it just because thats the way the sine function is? or is there a physics explanation to it?

Is it just because thats the way the sine function is? or is there a physics explanation to it?
The time derivative of sin(ωt) is ω·cos(wt), so when the argument of the sine function nears 90 degrees, the variaton of force with angle decreases to zero.

Bob S

ah i see, so its more of a maths explanation. thanks!

Excuse me, but I'd like to disagree with most of what has been said here, including the way question is posed, though I somewhat agree with Bob S's 1st response.

We can not talk about the force before we actually consider the properties of the SECOND particle or wire. We can talk about field potentials and its shape, but the *force* vectors will depend on the velocity and orientation of the charges in a second wire or 'test particle".

Therefore, when discussing the magnetic force, angles are result of compound interaction of at least TWO charges. The angle of a magnetic field potential is 90 degrees, i.e. magnetic field lines are circular (cylindrical in 3D time) and laying in a plane perpendicular to the velocity vector (wire length), while the magnetic FORCE, Lorentz force, is acting perpendicularly to both magnetic field and the velocity vector, but again, for this force is 'superimposed interaction', we need at least TWO of these.

In conclusion, use Lorentz force equation in vector format, so by applying cross product it would be more clear you are dealing with 90 degree angles. While Biot-Savart equation will describe field potential and how these field lines are circularly perpendicular to wire length.

Excuse me, but I'd like to disagree with most of what has been said here, including the way question is posed, though I somewhat agree with Bob S's 1st response.

We can not talk about the force before we actually consider the properties of the SECOND particle or wire. We can talk about field potentials and its shape, but the *force* vectors will depend on the velocity and orientation of the charges in a second wire or 'test particle".

Therefore, when discussing the magnetic force, angles are result of compound interaction of at least TWO charges. The angle of a magnetic field potential is 90 degrees, i.e. magnetic field lines are circular (cylindrical in 3D time) and laying in a plane perpendicular to the velocity vector (wire length), while the magnetic FORCE, Lorentz force, is acting perpendicularly to both magnetic field and the velocity vector, but again, for this force is 'superimposed interaction', we need at least TWO of these.

In conclusion, use Lorentz force equation in vector format, so by applying cross product it would be more clear you are dealing with 90 degree angles. While Biot-Savart equation will describe field potential and how these field lines are circularly perpendicular to wire length.

erm.. so how does what you said tie down with my question?

maybe i make it too complicated,

i was wondering why does the induced magnetic force on the current carrying wire seem to be linear at small angles but a curve at large angles.

i know the equation is cross product which means use sin.

but is it just because nature has it like that? so we use sin to describe this natural occurrence? or is there a physics explanation to this phenomenon?

erm.. so how does what you said tie down with my question?

maybe i make it too complicated,

i was wondering why does the induced magnetic force on the current carrying wire seem to be linear at small angles but a curve at large angles.

i know the equation is cross product which means use sin.

but is it just because nature has it like that? so we use sin to describe this natural occurrence? or is there a physics explanation to this phenomenon?
The vector cross product I x B in the Lorentz force is equal to the amplitude I·B of the product times the sine of the angle between the two vectors. The rate of increase of the sine slows down and stops when it reaches 90 degrees, than starts falling back to zero as the angle increases more.

Bob S

so its mainly a mathematical explanation of this natural occurrence and not so much a physics explanation as to why it behaves like that ?

Q: Angle between current and magnetic field?

In three dimensions there is more than one angle of interest. Please specify more precisely what angle are you talking about. All the naturally occurring angles here (built in as physical property) are 90 degrees. Any arbitrary angle you get is a result of relations of these right angles and your setup in terms of position and velocity vectors. This picture illustrates magnetic field of a single electron in a single instant, the magnetic field of all the electrons moving through time as electric current would look like a cylinder. Are you talking about any angle that has anything to do with the cuts or ends of this "cylinder"? If not, then the answer to this question is most likely to refer to the angle of 90 degrees, as illustrated.

Q: Why does the induced magnetic force on the current carrying wire seem to be linear at small angles but a curve at large angles.

y
| /z
|/
e----> x

Can you describe the angle you are talking about by using two electrons, please? Set one electron at the origin and give it some velocity along the x axis, as above, that will be our "current in a wire". The magnetic field of a single electron in a single instant will be parallel to Y and Z axis and perpendicular to X axis. I have no idea how this field is supposed to look in front and behind this yz plane, and if that is your question than I'd like to know the answer too.

Now, there will be no force unless there is at least one more electron (wire), so can you describe the location of this source of magnetic induction and its orientation in relation to above diagram so we know all the angles and especially the one you are talking about? Only then we will be able to say something about the forces.

Q: Is it just because nature has it like that, or is there a physics explanation to this phenomenon?

All the angles here that are "built-in" naturally are 90 degrees. You must be moving something, changing some orientation in relation to something else to get any other angles.

tiny-tim
Homework Helper
You could say it's mainly physical, because it's the amount of the field that is intersected, and if you tilt the circuit you'll intersect proportionately less of the field.

You could say it's mainly physical, because it's the amount of the field that is intersected, and if you tilt the circuit you'll intersect proportionately less of the field.

ok, so when the angle theta between the current carrying wire and the magnetic field is 0, (parallel), we have 0 induced force by the magnetic field on the wire

when it is 90 degrees, we have max force. sin(90) =1

so, my question is , why when i plot the graph of F vs this angle theta, the "increase" from theta=0 to aroung 50 is roughly linear.

but when theta reaches big angle values around 70 - 90, this "increase" slows down, meaning it forms a maximum point "n-shaped" curve, before peaking at the value 1, like a sine graph.

so is it because nature behaves this way, so we use the sine graph to represent nature, or is there a physics way to explain this phenomenon?

@tim: you are suggesting that the amount of field the wire cuts is proportional when you tilt it? ok.. but is there an explanation as to why higher angles cut lesser field while lower angles cut more field, as evident from the graph of F vs angle, which produces a sine-shaped graph.

tiny-tim
which in turn happens to be roughly a staright graph until about 50º. 