Magnetic effect of electric current and Biot-savart's Law or 'Ampere's Theorem'

 Concept of Field

It is a matter of common experience that for transmission of mechanical force, from one body to the other, a physical contact between them is necessary. There are cases where a body exerts a force on the other without any contact between them e.g., gravitational force or the electro-static force. This is termed as 'interaction at a distance'. To explain this interaction we take shelter of the concept of the field. Presence of a body, at any place, modifies the space around it. This modified space, around the body, is called a field. When two bodies are placed near each other, both the bodies have a field of their own. These fields interact with each other to give rise to a force. This is the reason why two charged bodies exert a force on each other or why two bodies, in this universe attract each other due to gravitational force. 

     Electro-static force arises due to interaction of the electric field while a magnetic force arises due to the interaction of two magnetic fields. If we place a magnetic pole in an electric field or an electric charge near a magnet no force is experienced by either of them. 

Magnetic effect of electric current - 'Oersted's Experiment'

In 1820, Oersted was able to show that an electric current flowing through a wire produces a magnetic field around it. 

     Consider a wire AB connected to a battery in such a way that electric current flows from A to B on closing the key (Fig. 1). Place a compass needle just below the wire AB. 

Fig. 1. Deflection of magnetic compass due to the magnetic field of a current. 

     Close the key 'K' so that a current starts flowing through the wire. It is observed that the compass needle gets deflected. Compass needle, which is a magnet, can only be effected by a magnetic field. The experiment indicates that a magnetic field gets developed around the wire when an electric current flows through it. The phenomenon is called magnetic effect of currents. 

Rules for Direction of Deflection of N-pole

The direction in which the North Pole of the compass needle is deflected can be determined by applying any one of the following rules :

(i) 'SNOW' rule. "If the direction of current flowing through the wire is from south (S) to north (N) and the wire is placed over (O) the needle, the North pole of the needle is deflected towards west (We). 

     If the wire is placed below the needle and the current still flows from south to north, north pole of the needle is deflected towards east. 

(ii) Ampere's rule. "Imagine a men to swim along the wire in the direction of the current with his face towards the needle so that the current enters at his feet and leaves at his head. The north pole of the needle will be deflected towards his left hand (Fig. 2).

Fig. 2. Ampere's rule. 

BIOT-SAVART'S LAW OR "AMPERE'S THEOREM" 

A wire carrying current has a magnetic field all around it. The intensity at any point, in this magnetic field can be obtained with the help of "Biot-Savart's law".

     Consider a small section dl, of a wire carrying q current 'i'. Let 'p' be the observation point at a distance 'r' from the centre 'O' of the element. The position vector of P subtends an angle θ with the direction of flow of current in the element dl as shown in Fig. 3.

Fig. 3. Strength of magnetic field, at any point due to a wire carrying current. 

     If  'dB' is a magnetic intensity at P due to this small element, it has been observed that

                 dB ∝ dl,   dB ∝ i

                 dB ∝ sin θ,  dB ∝ 1/r²

Combining all these factors, we get

                  dB ∝ i dl sin θ/r²

or              dB ∝ k × i dl sin θ/r²                  ... (1) 

where k is the constant if proportionality and its value depends upon the system of units selected. 

     Equation (1) gives the scalar form of Biot-Savart's law or Ampere's theorem. 

     In S.I. unit's, 'i' is taken in ampere, dl and r in metre. 

and      k =  μ0/4π   (where μ0 = 4π × 10-7 Wb A-1 m-1) 

dB = μ0/4π × i dl sin θ/r² × tesla.  

Vector form

     Biot-Savart's law, in vector form, can be written as 

               dB = μ0/4π × i × dl × r/r³ ... (2)

dB = μ0/4π × i × dl × r̂/r²                   ... (3) 

Note - (dB, dl r is in rightward arrow) 

     dl is a vector quantity whose length is proportion to the length of wire element and whose direction is given by the direction of current, r (rightward arrow) is a vector whose length is proportional to the straight line distance between 'O' and 'Q' (point of observation) and whose direction is along OQ. 

     According to the rule of cross product, direction of dB (rightward arrow) is perpendicular to the plane containing dl (rightward arrow) and r (rightward arrow). Fig. 4, OPQ is the plane containing dl (rightward arrow) and r (rightward arrow). Applying right hand thumb rule it can be concluded that 'dB' (rightward arrow) is perpendicular to OPQ and is directed upwards. 

Fig. 4. Direction of magnetic field. 

        |dB| (rightward arrow) = μ0/4π × i × dl × sin θ/r²

Case (i) if θ = 0°, |dB| = 0

i.e., there is no magnetic intensity at any point, on the axial line of the element. 

Case (ii) if θ = 90°,

|dB| (rightward arrow) = μ0/4π × i dl/r (maximum) 

Magnetic intensity at any point lying on a line perpendicular to the length of the element is maximum. 

Direction of lines of force :

Consider a long straight wire XY passing through the centre of a cardboard 'ABCD' and perpendicular to its plane. Spread some iron fillings on the cardboard. When an electric current is passed through the wire, the iron fillings arrange themselves around the wire in concentric circles, indicating that the lines of forces of magnetic field are in the form of concentric circles (Fig. 5). Lines of force due to two straight conductors, carrying currents in opposite directions are shown in Fig. 5(i) and Fig. 5(ii). The direction of lines of force is given by applying any of the following rules. 

Fig. 5. Magnetic field due to a long straight wire carrying current. 

     (i) Maxwell's cork screw rule. Imagine a right handed cork screw placed along the conductor carrying current with its axis coinciding with the direction of current so that if the screw is twisted it travels foreword in the direction of the current. The direction in which the ends of the handle turns gives the direction of lines of force (Fig. 6).

Fig. 6. Maxwell's cork screw rule. 

     (ii) Right hand thumb rule. Imagine the wire carrying current to be held in your right hand with its thumb pointing in the direction of electric current. The direction in which the fingers curl, gives the direction of lines of force if the magnetic field (Fig. 7).

Fig. 7. Right hand thumb rule. 

Magnetic field due to a long straight conductor carrying current

Consider an element XY of a long straight conductor AB carrying current i in the direction from A to B. Let People be the observation point at a distance x from the centre of the element. 

     Draw PM perpendicular to the length of conductor such that PM = r. 

     Let θ be the angle which PO makes with the direction of current  (Fig. 8).

     If dB be the magnetic intensity at P due to a current i through the element XY, then by Biot-savart's law, 

                dB = μ₀i/4π × dl sin θ/x²             ... (4) 

Fig. 8. Magnetic field at any point due to a long straight conductor carrying current. 

     In right angle ∆ OMP, θ + ∝ = π/2

or                   θ = π/2 - ∝

or            sin θ = sin (π/2 - ∝) 

or            sin θ = cos ∝

     Again, in right angle ∆ OMP

                cos ∝ = r/x

∴                    x = r/cos ∝

     Also  tan ∝ = l/r

∴                    l = r tan ∝

     Differentiating, we get

                    dl = r sec² ∝ d∝

     Substituting for dl, sin θ and x in equation (4), we get

         dB = μ₀/4π × i × (r sec² ∝ d∝) cos ∝/r² × cos² ∝

or     dB = μ₀i/4πr cos ∝ . d∝                        ... (5) 

     Applying right hand thumb rule, it can be concluded that direction of B will be perpendicular to the plane of paper directed outward. All other elements of the conductor will also produce intensity in the same direction. Net magnetic intensity at P due to whole of the conductor AB can be obtained by integrating equation (5) within the limits θ₁ and θ₂.

     ∴       B = B0∫ dB = μ₀i/4πr θ₂θ₁∫ cos cos ∝ d∝ 

       

                     = μ₀i/4πr [sin ∝]θ₂θ₁

or  B = μ₀i/4πr [sin θ₂ - sin θ₁] ... (6)

Special cases

     Case (i) When the point is situated symmetrically with respect to the two ends of the conductor :

Fig. 9. Point Of equidistant from two ends of conductor. 

     Let 'l' be the length of the conductor. 

     In this case, ∝₁ = − ∝ and ∝₂ = + ∝

     Using equation (6), we get

           B = μ₀i/4πr [sin ∝ - sin (- ∝)]

or       B = μ₀i/4πr × 2 sin ∝

     From Fig. 9, sin ∝ = l/√l² + 4r²

     ∴          B = 2 μ₀i/4πr × l/√l² + 4r²

or             B = μ₀i/2πr × l/√l² + 4r²

     Case (ii) If the conductor is of infinite length

     For an infinitely long conductor, angle θ₁ and θ₂ tends to - π/2 and + π/2 radian respectively. 

     Substituting θ₁ = - π/2 and θ₂

                                = + π/2 in equation (6) we get

          B = μ₀i/4πr [sin (π/2) - sin (- π/2)]

             = μ₀i/4πr [sin π/2 + sin π/2]

             = μ₀i/4πr [1 + 1] = μ₀2i/4πr

or     B = μ₀/2π × i/r ⇒B ∝ 1/r

 

Fig. 10.

 Key formulae

1. Biot-savart's law

         dB = μ0/4π × i dl sin θ/r² (Scalar form)

dB = μ0/4π × i × dl × r/r³ (Vector form)

(Note - dB, dl, r is in rightward arrow)

2. Field due to a long straight conductor carrying current

(i) Conductor of finite length

B = μ₀i/4πr [sin ∝₂ - sin ∝₁]

(ii) Conductor of infinite length

B = μ₀/2π × i/r