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Moving Charges and Magnetism

Contents:Magnetic field and origin of magnetic fieldUnit of magnetic fieldBiot-Savart lawApplication of Biot-Savart law: Magnetic field due to long straight conducting wire 4.1 Introduction:Magnetic field & Origin of Magnetic field: ● Magnetic field is a force field similar to that of Gravitational field and Electric field. ● Magnetic field is a region of space around a magnet within which if another magnet is brought, it experiences a force.It is a vector field and denoted by .● Electric field is created by electric charges (positive charges, negative charges or electric dipole) but in case of magnetism no such magnetic charges are proved to exist till date. Magnetic field is created by current. ● In a permanent magnet, magnetism is produced by orbital and spinning motion of electrons in an atom. Motion of electrons in an atom constitutes a circulating current, which creates magnetic field within the magnetic material.● Fundamental/basics origin of magnetism is a magnetic dipole or current carrying loop/circulating current. Units of Magnetic field: SI units:Tesla (T) or

(\begin{array}{c} \frac{N}{A . m .} \end{array})

. Magnetic field

\vec{B}

is sometime called “Magnetic flux density” and hence it has another unit,

(\begin{array}{c} \frac{w e b e r}{m^{2}} = \frac{W b}{m^{2}} \end{array})

CGS units: Gauss (G); 1 T = 10

^{4}

G 4.2 Biot-Savart Law After Oersted’s experiment had concluded that a current carrying conductor produces magnetic field around it, French physicists Jean Baptiste Biot and Felix Savart deduced an empirical law to determine the magnitude and the direction of the magnetic field that a current element may produce at any point in space around it. Current element: Current element is a small length segment of a current carrying conductor. It is a vector quantity having magnitude

∣ I d \vec{l}

∣

and direction along the tangent drawn at that segment in the direction of current (shown in the figure-4.1). Fig.-4.1: Biot-Savart Law and Current Element According to Biot-Savart law i.

d B \propto I

ii.

d B \propto d I

andiii.

d B \propto

\frac{1}{r^{2}}

Combining all one gets,

d B \propto

\frac{I d l}{r^{2}}

. In vector form the law is written as

d \vec{B}

= \frac{𝜇_{0}}{4 𝜋} \frac{(I d \vec{l} \times \vec{r})}{r^{3}}

, where

\frac{𝜇_{0}}{4 𝜋}

is the proportionalityconstant having value

\frac{𝜇_{0}}{4 𝜋} = 10^{- 7} T m A^{- 1} (o r W b m^{- 1} A^{- 1}) .

Scalar form of the complete formulato be used while solving numerical problems is

d B = \frac{𝜇_{0}}{4 𝜋} \frac{(I d l s i n 𝜃)}{r^{2}}

Fig.-4.2: Right-hand Thumb Rule

Direction of dB: The direction of the dB is in the direction of

\vec{d l} \times \vec{r},

which can be determined easily by using ‘Right-hand thumb Rule’ for cross product. ‘Right-hand thumb Rule’ for cross product has been demonstrated in the figure-4.2: Biot-Savart law can be used to determine magnetic field produced by a current conductor of any arbitrary shape considering that to be made of many small current element.

d B = \frac{𝜇_{0}}{4 𝜋} \frac{(I d l s i n 𝜃)}{r^{2}}

suggests that the magnetic field is zero at a point on the axis of the current element as

𝜃 = 0

over there. The magnetic field due to current element will be the maximum at any point in the plane passing through the current element and perpendicular to the axis of the current element as

𝜃 = 90^{°}

4.3 Application of Biot-Savart Law Magnetic field due to long straight current conductor: Magnetic field produced by a long straight wire can be determined by using Bio-Savart law.Let us consider a long straight wire PQ (shown in the figure-4.3) carrying a current ; we wish to calculate magnetic field at M point created by the wire. Here

a

is the perpendicular distance from the point M to be to the conducting wire PQ.

𝜃_{1}

is measured from the perpendicular (MN) to the down and is measured in the upward direction from the perpendicular (MN). To calculate magnetic for the wire first we consider a small current element

\vec{d l}

and position vector of the point M is

\vec{r}

. According to Biot-Savart law the magnetic field at M,

d \overset{}{\vec{B} = \frac{𝜇_{0}}{4 𝜋} \times \frac{I (d \vec{l} \times \vec{r})}{r^{3}} \Rightarrow d B = \frac{𝜇_{0}}{4 𝜋} \times \frac{I d l \sin 𝜙)}{r^{2}}}

.................................................(4.1). From

𝛥 M O N,

𝜃 + 𝜙 = 90 °

\Rightarrow 𝜙 = (90 ° - 𝜃)

∴ \sin 𝜙 = \sin (90 ° - 𝜃) = \cos 𝜃

Now,

\cos 𝜃 = \frac{a}{r} \Rightarrow r s e c 𝜃

Also

\tan 𝜃 = \frac{l}{a} \Rightarrow a \tan 𝜃 \Rightarrow d l = a s e c^{2} 𝜃 d 𝜃

..............................................................(4.2) Putting the value of

d l

and

r

into equation (4.1) we get,

d B = \frac{𝜇_{0}}{4 𝜋} \times \frac{I (a s e c^{2} 𝜃 d 𝜃) \cos 𝜃}{a^{2} s e c^{2} 𝜃 d 𝜃} \Rightarrow d B = \frac{𝜇_{0} I}{4 𝜋 a} \cos 𝜃 d 𝜃

..........................................(4.3) To get the magnetic field created by the full wire one needs to integrate the equation (4.3) between the limit

- 𝜃_{1}

and

- 𝜃_{2}

that is

B = \int_{- Q_{1}}^{Q_{2}} d B = \frac{𝜇_{0} I}{4 𝜋 a} \int_{- Q_{1}}^{Q_{2}} \cos 𝜃 d 𝜃 = \frac{𝜇_{0} I}{4 𝜋 a} [\sin 𝜃_{1} + 𝜃_{2}] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.4)

Equation (4.4) gives the magnetic field produced by a finite wire in terms of the angles (

𝜃_{1}

and

𝜃_{2}

) subtended at the points of observation by the ends of the wire. ● If the wire, PQ is infinitely long and the point M is in the middle then ●

𝜃_{1} = 𝜃_{2} = \frac{𝜋}{2} a n d h e n c e B = \frac{𝜇_{0} I}{2 𝜋 a}

● If the wire, PQ is infinitely long and the point M is near Q then●

𝜃_{1} = \frac{𝜋}{2} a n d 𝜃_{2} = 0 a n d h e n c e B = \frac{𝜇_{0} I}{2 𝜋 a}

● Magnetic field at any end of an infinitely long conducting wire is half of the value of the magnetic field at the middle i.e.,●

B = \frac{1}{2} (\begin{array}{c} \frac{𝜇_{0} I}{2 𝜋 a} \end{array})

Moving Charges and Magnetism

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