Moving Charges And Magnetism

Magnetic field due to current element, Biot-Savart; Magnetic field on the axis of a circular current loop

• Static charges produce an electric field while current or moving charges produce magnetic field (B).

• Magnetic field of several sources is the vector addition of magnetic field of each individual source.

Lorentz Force

• Consider a point charge q moving in the presence of both electric and magnetic fields.

Let

q − Point charge

v − Velocity of point charge

t − Time

r − Distance

B (r) − Magnetic field

E (r) − Electric field

∴ Force on the charge, = F_electric + F_magnetic

This force is called Lorentz force.

• Force due to magnetic field depends on q, v, B. Force on negative charge is opposite to that of positive charge.

• Magnetic force is a vector product of velocity (v) and magnetic field (B). It vanishes, if v and B are parallel or anti-parallel.

• Magnetic force is zero, if charge is not moving.

• Unit of magnetic field (B) is tesla (T).

Magnetic Force on a Current Carrying Conductor Placed in Magnetic Field:

A straight rod carrying current is considered.

Let

A − Cross-sectional area of the rod

l − Length of the rod

n − Number density of mobile charge carriers

I − Current in the rod

v_d − Average drift velocity of mobile charge carrier

B − External magnetic field

Force on the carriers,

F = (nAl) qv_d × B

Since current density, j = nqv_d

∴ F = [(nqv_d)Al] × B

F = [jAl] × B

F = I l × B

Where,

l is the vector magnitude of length of the rod

• For a wire of arbitrary shape,

The magnetic effect of current was first discovered by H.C. Oersted. He observed that the flow of charges in a conductor produces magnetic effect around it. He demostrated this phenomenon using the experimental set-up shown below.

When the current in the circuit was anticlockwise, the magnetic needle was found to deflect towards West. The needle deflected towards East when the direction of current was reversed. This experiment proved that current has magnetic effect associated with it. 

The Biot–Savart Law

Let:

XY = Current-carrying conductor

I = Current in the conductor

dl = Infinitesimal element of the conductor

dB = Magnetic field at point P

r = Distance of point P from the element

According to the Biot–Savart law, the magnetic field is proportional to the current and element length and inversely proportional to the square of the distance.

That is,

dB ∝

=

Here,

= Constant of proportionality = 10⁻⁷ Tm/A

μ₀ = Permeability in free space

Magnetic field on the axis of a circular current loop

I = Current in the loop

R = Radius of the loop

X-axis = Axis of the loop

X = Distance between O and P

dl = Conducting element of the loop

• According to the Biot–Savart law, the magnetic field at P is

dB =

r² = x² + R²

|dl × r| = rdl      (Because they are perpendicular)

 

• dB has two components: dB_x and dB_⊥. dB_⊥ is cancelled out and only the x-component remains.

∴ dB_x= dBcos θ

cos θ =

∴ dB_x =

• Summation of dl over the loop is given by 2πR.
  ∴ B = =

• For the magnetic field at the centre of the loop, x = 0.

∴

Right-Hand Thumb Rule
Maxwell’s right-hand thumb rule indicates the direction of magnetic field if the direction of current is known.
According to this rule, if we grasp the current-carrying wire in our right hand such that our thumb points in the direction of the current, then the direction in which our fingers encircle the wire will tell the direction of the magnetic field lines around the wire.



When the thumb points upwards, the curled fingers are anticlockwise. So, the direction of the magnetic field is anticlockwise.
When the thumb points downwards, the curled fingers are clockwise. So, the direction of the magnetic field is clockwise.

• When a charged particle having charge q moves inside a magnetic field with velocity v, it experiences a force .

• When is perpendicular to, the force on the charged particle acts as the centripetal force and makes it move along a circular path.

• Let m be the mass of charged particle and r be the radius of the circular path.

           Then $q\left(\overrightarrow{v}\times \overrightarrow{B}\right)=\frac{m{v}^2}{r}$

        v and B are at right angles

      $$\begin{gathered} \therefore qvB=\frac{m{v}^2}{r} \\ \Rightarrow r=\frac{mv}{Bq} \end{gathered}$$

• Time period of the circular motion of a charged particle is given by

            

        ⇒  

• Angular frequency,

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