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I am trying to understand what a magnetic field vector tells us about the magnetic field. I understood that a vector is just a representation of a point and how much it is moved in x,y and z direction from its origin.

But how can this explanation apply to the magnetic field?

source: http://www.cprogramming.com/tutorial/3d/rotationMatrices.html second paragraph

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Why We need vectors?

Vectors represent a set of physical quantities that require both it's magnitude (or length) and direction (spatial orientation) to describe the quantity completely. Take a real life situation as an example. You ask me how many marbles I have in my bag. I say it's 10. Tat"10" is sufficient enough for the answer. Now, ask another question. How far is your house from here? You would probably say about 20 km; not just 20. The additional "km" have to be included in order to specify the solution completely. Now, we ask, how to get to your house? Of course you need the aid of north, south, east, west directions.

Likewise is certain physical quantities. Mentioning just their number alone seems to be out of sense. We call these quantities vectors. These quantities require the direction (represented by arrows) along each coordinate plus the value assumed at some point, represented by the space point.

Is magnetic field a vector?

The answer is a no. It's actually a pseudovector. Consider a straight long current carrying wire and a steady current flowing through that wire. This will generate a magnetic field, with the field lines forming closed loops or concentric rings about the wire. How to specify the direction of a closed loop. The field lines are not even circulating actually. Then how we know that the field lines have a curl?

How magnetic field became a vector?

Do the same experiment as mentioned above. At first, make the current flow upwards through the wire. It will generate a magnetic field at some point near the wire, whose direction can be measured by a compass. Keep the position of the compass fixed (i.e., the distance between the wire and the compass box). The compass needle deflects in a direction of the field line at that point (which we represent by drawing a tangent to the loop at that point). Now reverse the current. Surprisingly, you see that the compass needle deflects in the opposite direction. What does it mean? This means, even though the field lines are not circulating, their direction has to be specified to indicate the direction of flow of current and thereby the force acting on a moving charge. So we assumed tat if a positive current move upwards through a wire, the magnetic field lines curl anti-clockwise. If it flows downwards, the field lines curl clockwise.

This is why we say the magnetic field is a pseudovector.

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A vector field is a function which associates to a point of the space (domain) a vector, namely a set of number. So, in this case, a well defined magnetic field vector gives you these sets of numbers for every point of the space. Each of these sets of numbers contains information about the intensity and the direction of the magnetic field in the relative point so that you can evaluate the effect it has on a charged particle which passes in that point. The way you can do that is by applyng Maxwell'equation. From those you can derive the Lorentz force: $$ \mathbf{F}=q\mathbf{v}\times\mathbf{B} $$ where $\mathbf{F}$ is the force acting on particle, $q$ is the electric charge, $\mathbf{v}$ its velocity and $\mathbf{B}$ the magnetic vector field. If you also know the mass $m$ then you can study the motion of the particle by integrating the second Newton's law. The vector which correspond to your description is the position $\mathbf{r}$ of the particle whose second time derivative is just the acceleration $\mathbf{a}$.

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Generally, a magnetic field is produced either by a moving charge or a magnet. This magnetic field's strength and direction at a point are described by magnetic field vector.

The closer we move the point to magnet, the denser is the field at the body. Thus, the length of magnetic field vector increases. magnetic field vector is also used in determining the force on a moving charge.

$$\vec{F} = q(\vec{V} \times \vec{B})$$ where q = charge in coulumb, v = velocity of the charge B = magnetic field vector.

In this way, there are many applications of magnetic field vector.

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A vector is a quantity having a direction and a magnitude.

Magnets have a magnetic influence, which can be visualized as a physical vector field.

A magnetic field vector allows you to predict the influence of a magnet on a magnetic material.

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  • $\begingroup$ Can I deduce from your explanation that if I calculate the terrestrial magnetic field vector on a certain point in space, I know: how much lorentz force there is on this spot, in which direction the terrestrial magnetic field is going, and one more thing? is this correct? $\endgroup$ – ohiliouh Apr 29 '15 at 12:59
  • $\begingroup$ @JaqcuesMartin: A Lorentz force arises from interaction between a magnetic and an electric field. You would also have to know the electric field vector at that certain point in space in order to calculate the Lorentz force. A Lorentz force occurs on a point charge, so you would have to know the velocity and charge of the point charge as well. $\endgroup$ – Vatsal Manot Apr 29 '15 at 13:02

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