Magnetic Field

A magnetic field is produced by moving charges or current on its surrounding. It exerts force, \vec F to other charges moving or present in the field. Magnetic field is a vector quantity with symbol \vec B. It can be described by a magnitude and direction. The direction of a magnetic field depends on the direction of a compass needle would point and the magnitude depends on the strength of the magnetic force.

A Danish physicist named Hans Christian Oersted discovered the basic principle of electromagnetism which states that:

"Whenever electrons move through a conductor, a magnetic field is created in the region around the conductor."

Magnetic Field and Magnetic Forces on Moving Charges

1. The magnitude of the magnetic force is directly proportional to the magnitude of the charge. Two charges of different magnitude moving with the same velocity on the same field experience different amount of magnetic force. The one with greater charge would likely to experiences greater force that the one with smaller charge. For example, if 1 C and 2 C of charge is moving around the same magnetic field at the same velocity, the 2 C charged particle experiences twice as much as the force experienced by the 1 C charged particle.

1. The magnitude of the force varies directly as the magnitude or strength of the field. Magnetic force is stronger on stronger magnetic field. In the case of broken bar magnet, the smaller piece would have weaker magnetic field, thus, weaker magnetic force, while the longer field would have strong magnetic force and field. For example, if the magnitude of the field is doubled, like using two magnets instead of one, with constant charge and velocity, the magnetic force will also double.

1. The magnetic force depends on the particle’s velocity. The faster a charged particle moves, the stronger the magnetic force applied to it. Slower charged particle experiences lesser force. A charged particle at rest does not experience magnetic force.

1. The direction of the magnetic force is always perpendicular to the direction of both the magnetic field and velocity. It does not have the same direction as the magnetic field. The magnitude of the magnetic force is proportional to the component of the velocity perpendicular to the field. When the component of the velocity is zero, the force is also zero. The magnitude of the force can be expressed mathematically as

$$F = |q|vB = |q|v_{\perp} B \sin \phi$$     (1)

where $$|q|$$ is the magnitude of the charge and $$\phi$$ is the angle measured between the direction of $$\vec v$$ and the direction of $$\vec B$$.

 When a charge is moving parallel to a magnetic field, the magnetic force it experiences is zero. When a charge is moving at an angle $$\phi$$ to a magnetic field, it experiences a magnetic force whose magnitude is $$F = |q|vB = |q|v_{\perp} B \sin \phi$$. When a charge is moving perpendicular to a magnetic field it experiences a maximal magnetic force whose magnitude is $$F_{max}=qvB$$.

To find the direction of the magnetic force, we use the right-hand rule. First, draw the vectors  $$\vec v$$ and $$\vec B$$, tail to tail. Suppose your palm is the $$\vec v-\vec B$$ plane. Now, imagine turning $$\vec v$$ to the direction of $$\vec B$$. Considering a line perpendicular to the plane, curl your fingers around the line so that there’s a rotation from $$\vec v$$ to $$\vec B$$. The direction to which the thumb points pertains to the direction of the force on a positive charge. Below is the visual representation of the use of right-hand rule in determining the direction of the magnetic force.

 Right-hand rule for the direction of magnetic force on a positive charge moving in a magnetic field. Right-hand rule for the direction of magnetic force on a negative charge moving in a magnetic field.

This implies that the magnitude and direction of the force $$\vec F$$ exerted on a charge q moving with velocity $$\vec v$$ in a magnetic field $$\vec B$$ is given by the equation

$$\vec F = q\vec v \times \vec B$$     (2)

The above equation is valid for all moving charges, either positive or negative. The difference is that the direction of $$\vec F$$ if q is negative is opposite to that of $$\vec v \times \vec B$$.

When two oppositely charged particles of the same magnitude and velocity is moving at the same magnetic field, both will experience the same amount of magnetic force but with opposite direction.

Note that equation (1) gives the magnitude of the magnetic force and that $$\phi$$ is the angle between the direction of $$\vec v$$ and $$\vec B$$. We may take $$B \sin \phi$$ as the component perpendicular to $$\vec v$$ denoted as $$B_{\perp}$$. We can now write (1) as

$$F = |q|vB_{\perp}$$

The unit for the magnitude of the magnetic field B is tesla abbreviated as T. It is named after the prominent Serbian-American scientist and inventor Nikola Tesla.

$$1\; T = 1\; N/A \cdot m$$