# Magnetic Effect of Current Part -3

Torque on a Conductor:

Torque on Current Carrying Coil in Magnetic Field:

A magnetic dipole is the limit of either a closed electric circuit or a pair of poles because the size of the source is reduced to zero keeping the magnetic moment constant. It is now shown that a constant current I through a rectangular loop placed in a uniform field has torque. It has no mains power. This behavior is similar to an electric dipole in a uniform electric field.

Consider the case of placing a rectangular loop such that the uniform magnetic field B is in the plane of the loop. The field exerts no force on her two arms PS and QR in the loop. It is perpendicular to arm PQ of the loop and exerts a force F1 on arm PQ directed in the plane of the loop. Its size is

F1=IzB

Similarly, force F2 is applied to arm RS and F2 is directed out of the plane of the paper.

F2 =IzB= F1

Therefore the net force on the loop is zero. A torque acts on the loop because the two forces F1 and F2 cancel each other. Here we can see that the torque applied to the loop tends to cause it to rotate counterclockwise.

τ = F1 $\left(\frac{Y}{2}\right)$ + F2 $\left(\frac{Y}{2}\right)$

= Izb$\left(\frac{Y}{2}\right)$ + Izb$\left(\frac{Y}{2}\right)$

= I(yz) B

= IAB ……. (1)

Where A = y × z is the area of ​​the rectangle.

Case 2

Now consider the case where the plane of the loop is not aligned with the magnetic field, but at an angle with it. And consider the angle between the magnetic field and the normal to the coil as the angle Θ.  The forces on the two arms, QR and SP, are equal and opposite, acting along the axis of the coil connecting the centers of mass of QR and SP. Since they are collinear along the axis, they cancel each other out and produce no net force or torque. The forces on arms PQ and RS are F1 and F2. Moreover, they are also of the same and opposite magnitude,

F1 = F2 = IzB

Since they are not collinear, they form a pair. However, the effect of torque is less than if the plane of the loop passed along the magnetic field. The magnitude of the torque on the loop is:

τ = F1 $\left(\frac{Y}{2}\right)$  sinθ F2 $\left(\frac{Y}{2}\right)$ inθ

= I(y×z) B sin θ

= IABsinθ ……. (2)

Therefore, the torque in equations (1) and (2) can be expressed as the vector product of the magnetic moment of the coil and the magnetic field. Therefore, the magnetic moment of the current loop can be defined as

m = IA

where A is the direction of the area vector. If the angle between m and B is θ, equations (1) and (2) can be expressed as follows.

τ = m × B

where m is the magnetic moment and B is the uniform magnetic field.

Magnetic Dipole Moment:

The magnetic moment is a quantity that describes the magnetic strength and orientation of a magnet or other object that produces a magnetic field. More specifically, magnetic moment refers to the magnetic dipole moment, i.e. the component of the magnetic moment that can be represented by a magnetic dipole. A magnetic dipole is a magnetic north pole and a magnetic south pole separated by a small distance.

The magnetic dipole moment is the current and area or energy divided by the magnetic flux density. The dipole moment has units of meters, kilograms, seconds, amperes, and amperes squared. The unit of the centimeter-gram-second system is erg (unit of energy)/gauss (unit of magnetic flux density). 1,000 ergs per gauss equals 1-ampere square meter.

Magnetic Moment formula:

Magnetic dipole moment – ​​The magnetic field B due to a current loop carrying current i of radius R separated by a distance l along the axis is given by

Now, considering points so far from the current loop that l>>R holds, we can approximate the field as

Now the area of ​​the loop is A

So the magnetic field can be written as

We can write this new quantity μ as a vector pointing along the magnetic field.

Notice the striking resemblance to the electric dipole field.

Unlike the electric field, the magnetic field has no "charge" or "counterpart". In other words, the magnetic field has no source or sink, only dipoles can exist. Anything that can generate a magnetic field has both a source and a sink. H. There are both north and south poles. In many respects, magnetic dipoles are fundamental entities capable of creating magnetic fields.

Most elementary particles behave essentially like magnetic dipoles. For example, the electron itself behaves like a magnetic dipole and has a magnetic spin dipole moment. This magnetic moment is inherent because the electron has no region A (it is a point body) and does not rotate around itself, but it is fundamental to the nature of the electron's existence.

The 'N' times magnetic moment of a wire loop can be generalized as

μ = NiA

The field lines of the current loop resemble those of an ideal electric dipole.

If you've ever split a magnet in two, you'll know that each piece forms a new magnet. The new pieces also include the north and south poles. It seems impossible to get just the North Pole.

Current Loop as a Magnetic Dipole:-

Ampere found that the distribution of magnetic lines of force around a finite current-carrying solenoid is similar to that produced by a bar magnet. This is evident from the fact that a compass needle when similar deflections moved around these two bodies. After noting the close resemblance between these two, Ampere demonstrated that a simple current loop behaves like a bar magnet and put forward that all the magnetic phenomena are due to circulating electric current. This is Ampere’s hypothesis.

The magnetic induction at a point along the axis of a circular coil carrying current is,

B = µonI$\frac{{a}^{2}}{2}$(a2+x2)$\frac{3}{2}$

The direction of this magnetic field is along the axis and is given by the right-hand rule. For points that are far away from the center of the coil, x>>a, a2 is small and it is neglected. Hence for such points,

B = µonI$\frac{{a}^{2}}{2}$x3

If we consider a circular loop, n = 1, its area A = πa2

So, B = µo I$\frac{{a}^{2}}{2}$πx3

The magnetic induction at a point along the axial line of a short bar magnet is

B = $\left(\frac{{\mu }_{o}}{4\pi }\right)\left(\frac{2M}{x3}\right)$

Or, B = $\left(\frac{{\mu }_{o}}{2\pi }\right)\left(\frac{M}{x3}\right)$

Comparing equations (1) and (2), we find that

M = IA

Hence a current loop is equivalent to a magnetic dipole of moment M = IA

The magnetic moment of a current loop is defined as the product of the current and the loop area. Its direction is perpendicular to the plane of the loop.

Magnetic Dipole Moment of a Spinning Electron:

According to Neil Bohr's atomic model, a negatively charged electron revolves around a positively charged nucleus in a circular orbit of radius r. An electron rotating in a closed path generates an electric current. The counterclockwise movement of the electron produces a clockwise current in the conventional current.

Since the electron is negatively charged, the normal current flows in the opposite direction to its movement. The magnetic moments associated with the orbital motion of two electrons are shown in the figure. An electron rotating in an orbit of radius 'r' corresponds to a magnetic layer with a magnetic moment of 'M'

M = i\ times A

Here "i" is the current corresponding to the spinning electron and A is the area of ​​the orbit. Where '\tau' is the spin time of the electron.

i = charge/period = e/\tau = $\frac{e}{\frac{2\pi }{\omega }}$

or i = $\frac{e\omega }{2\pi }$

Here 'ω' is the angular velocity of the electron.

So M =  $\frac{e\omega }{2\pi }$\ times (πr2) =

According to Bohr's theory, an electron can move in an orbit where its angular momentum is an integral multiple of h/2π, where "h" is Planck's constant.

mr2ω = n $\left(\frac{h}{2\mathrm{\pi }}\right)$

Or r2ω = n $\left(\frac{h}{2m\mathrm{\pi }}\right)$

Substituting r2ω in the equation, we get,

M = n $\left(\frac{eh}{\mathrm{\pi m}}\right)$

Since an atom can have a large number of electrons, the magnetic moment of an atom is the vector sum of the magnetic moments of the different electrons.  The term $\left(\frac{eh}{\mathrm{\pi m}}\right)$ is called the Bohr magneton. This is the smallest value of the magnetic moment of the electron. Each atom can have a magnetic moment that is a fixed multiple of the Bohr magneton. Thus, the magnetic moment is also quantified at the atomic and subatomic level.

In addition to the magnetic moment due to orbital motion, the electron has a magnetic moment due to its spin. Thus, the resultant magnetic moment of an electron is the vector sum of its orbital magnetic moment and its spin magnetic moment.

Galvanometer:

A galvanometer is a device used to detect or measure the magnitude of a small electric current. Current and its intensity are usually expressed by the movement of a magnetic needle or coil in a magnetic field, which is an essential part of a galvanometer.

Since its discovery in the 19th century, the galvanometer has seen many iterations. Some of the different types of galvanometer are tangent galvanometer, astatic galvanometer, mirror galvanometer, and ballistic galvanometer. However, the main type of galvanometer widely used today is the D'Arsonval/Weston type or the moving coil type. A galvanometer is essentially a historical name given to a moving coil electric current detector.

What is a moving coil galvanometer? A moving coil galvanometer is a device used to measure electric currents. It is a sensitive electromagnetic device that can measure small currents down to a few microamperes.

Moving coil galvanometers are mainly divided into two types.

Hanging galvanometer

Spiral Coil or Weston Galvanometer

Principle of moving coil galvanometer

A current-carrying coil, when placed in an external magnetic field, experiences a magnetic moment. The angle through which the coil is deflected by the magnetic moment is proportional to the magnitude of the current in the coil.

Construction and Diagram of Moving Coil Galvanometer

A moving coil galvanometer consists of a rectangular coil of several turns, usually made of thin insulation or fine copper wire wound on a metal frame. The coil is free to rotate about a fixed axis. A phosphor-bronze strip connected to a movable torsion head is used to suspend the coil in a uniform radial field.

The important characteristics of the material used to hang the coil are conductivity and a low value of torsional constant. To improve the strength of the magnetic field and direct the field radially, a cylindrical soft iron core is placed symmetrically inside the coil. The lower part of the coil is attached to a phosphor-bronze spring with a small number of turns. The other end of the spring is connected to the mounting screws.

Operation of Moving Coil galvanometer

A current I passes through a rectangular coil of n turns and cross-sectional area A. If this coil is placed in a uniform radial magnetic field B, a torque τ is applied to the coil.

Let us first consider one turn of a rectangular coil ABCD of length l and width b. It is suspended in a magnetic field of strength B so that the plane of the coil is parallel to the magnetic field. Since the sides AB and DC are parallel to the direction of the magnetic field, they have no effective force under the influence of the magnetic field. The sides AD and BC are perpendicular to the direction of the field, affecting the effective force F, with F = BIl

Using Fleming's left-hand rule, we can determine that the forces AD and BC are opposite. When equal and opposite forces F, called a couple, act on the coil, it produces a torque. This torque causes the coil to bend.

We know that torque τ = force x perpendicular distance between forces

τ = F × b

Substituting the value of F that we already know,

The torque τ acting on the single-circuit winding ABCD = BIl × b

where lx b is the coil area A,

Thus, the torque acting on n turns of the coil is given by the formula

τ = nIAB

The magnetic torque produced in this way causes the coil to rotate and the phosphor bronze strip to twist. The spring S attached to the coil in turn produces a countermoment or restoring torque kθ, resulting in a uniform angular deflection. In balance,

kθ = nIAB

Here, k is called the torsional constant of the spring (returns a couple per unit of rotation). The deflection or twist θ is measured as a value indicated on the scale by a pointer attached to the suspension wire.

θ= (nAB/k)I

Therefore, θ I

The quantity nAB/k is constant for a given galvanometer. So it is clear that the deflection of a galvanometer is directly proportional to the current passing through it.

Solved Question: What is the introduction of a cylindrical soft iron core into a moving coil galvanometer?

Solution: A cylindrical soft iron core placed inside the galvanometer increases the strength of the magnetic field and thus improves the sensitivity of the instrument. It also makes the field radial so that the angle between the plane of the coil and the magnetic lines of force remains zero during the rotation of the coil.

Sensitivity of a moving coil galvanometer

A general definition of the sensitivity experienced by a moving coil galvanometer is given as the ratio of the change in galvanometer deflection to the change in coil current.

S = d9/dl

The sensitivity of the galvanometer is greater if the device shows a greater deflection at a lower value of current. There are two types of sensitivity namely current sensitivity and voltage sensitivity.  Current sensitivity

The deviation θ per unit current I is called the current sensitivity θ/I

θ/I = nAB/k

Voltage sensitivity:

The deviation θ per unit of voltage is called voltage sensitivity θ/V. Dividing both sides by V in the equation θ= (nAB / k)I

θ/V= (nAB /V k)I = (nAB / k) (I/V) = (nAB /k) (1/R)

R represents the effective resistance of the circuit.

It is worth noting that voltage sensitivity = current sensitivity/winding resistance. Therefore, if R remains constant, voltage sensitivity    current sensitivity.

Galvanometric Usefulness Profile

It is the ratio of the full-scale deflection current to the number of scales on the instrument scale. It is also the inverse of the current sensitivity of the galvanometer.

Factors affecting the sensitivity of the galvanometer

a) Number of coil turns

b) Area of ​​coil

c) Strength of magnetic field  B

d) Couple size per torsional unit  k/nAB

Applications of Galvanometer

A moving coil galvanometer is a very sensitive device which allows it to be used to detect current in any  circuit. When a galvanometer is connected to a Wheatstone bridge circuit, the pointer of the galvanometer shows zero deflection, i.e. no current passes through the device. The cursor will tilt left or right depending on the direction of the flow.

A galvanometer can be used for measurement

a) The value of the current in the circuit connecting it in parallel with a small resistance.

b) Voltage by connecting it in series with high resistance.

Convert Galvanometer to Ammeter:

A galvanometer is converted into an ammeter by connecting it in parallel with a small resistance called a shunt resistance. A suitable shunt resistor is selected according to the range of the ammeter.

In the given circle

RG - Galvanometric resistance

G - Galvanometer coil

I - total current through the circuit

IG - total current through the galvanometer equal to the full scale reading

Rs - Shunt resistance value

When the current IG passes through the galvanometer, the current through the shunt resistor is given by the formula IS = I - IG. The voltages across the galvanometer and the shunt resistor are the same because of the parallelism of their connection.

Hence RG.IG= (I-IG).Rs

The value of S can be obtained using the above equation.

Convert a Galvanometer to a Voltmeter:

A galvanometer is converted into a voltmeter by connecting it in series with a large resistance. A suitable high resistance is selected according to the range of the voltmeter.

In the given circle

RG = galvanometric resistance

R = high resistance value

G = Galvanometer coil

I = total current through the circuit

IG = total current through galvanometer corresponding to full-scale deflection

V = voltage drop in series connection of galvanometer and high resistance

When a current IG passes through a series combination of a galvanometer and a large resistance R, a voltage drop is obtained in the branch ab

V= RG.IG R.IG

The value of R can be obtained using the above equation.

Solved question: A moving coil galvanometer with a resistance of 100 Ω is used as an ammeter with a resistance of 0.1 Ω. The maximum deflection current of the galvanometer is 100 μA. Find the current in the circuit so that the ammeter shows maximum deflection.

Solution: Given that RG = 100 Ω, Rs = 0.1 Ω, IG = 100 μA

We know that RG.IG= (I- IG).RS

Therefore I = (RG . IG IG.Rs)/RS

I = (1 RG/RS). IG

If the given values ​​are substituted, we get I= 100.1mA