Magnetic Effects of Electric Current: Meaning, Magnetic Flux, Solved Examples

Magnetic Effects of Electric Current: We all know about the electromagnets or motors which work on electricity. But how does the electromagnet produces a magnetic field? And if the electricity is related to the magnetic field, how can we produce a magnetic field using the current. Do all the wires in our home also produce the magnetic field?

What are the laws which govern these phenomena? Will the compass work properly when near a current-carrying wire?
What is a solenoid, toroid? What is ampere’s circuital law? How to determine the direction of the induced magnetic field and induced current using the fleming rule? How to calculate magnetic field due to different shapes of the conductor? Let us read further to understand more.

What is a Magnetic Field?

A static charge produces an electric field only, but both the electric field and magnetic field are produced when a charge is moving. We can say that a moving charge produces a magnetic field. When an electric current is flowing inside a conductor, the electrons in the conductor are in motion. Due to the motion of this electron, a magnetic field is generated around the conductor. Magnets also generate magnetic fields.

Oersted’s Experiment

Hans Christian Oersted was one of the leading scientists of the nineteenth century. He accidentally discovered that a compass needle gets deflected near a current-carrying wire and later experimented with showing that magnetism and electricity are related. His research laid the foundation for modern technologies such as television, radio, etc.
When a compass needle is brought near a current-carrying wire, the compass’s needle gets deflected, and as we bring the compass closer to the wire, the deflection of the needle increases.

Fig: When the Current Passes Through the Wire Above the Compass, the Compass Needle gets Deflected.

Magnetic Field Lines

Fig: Magnetic Field Lines in Different Types of Magnets.

Magnetic field lines are imaginary lines that are used to describe the magnetic field in space pictorially. Thus a magnet will have magnetic field lines present around it, and if a magnetic substance or a magnet is present in the space, it will experience some force.
Properties of the magnetic field are as follows:

1. Magnetic field lines always form a loop.This is due to the reason that magnetic monopoles do not exist.
2. The magnetic field lines emerge from the north pole and enter through the south pole from a magnet.
3. Two magnetic field lines never intersect. Magnetic field lines do not intersect because it represents the direction of the net magnetic field. If it intersects, then there will be two directions for the net magnetic field at the point of intersection, which is impossible.

Magnetic Flux

Magnetic flux can be defined as the amount of magnetic field passing through an area. It represents both the direction and the magnitude of the magnetic field passing through that area. It is denoted by using the Greek letter Phi\((\phi)\). Its SI unit is Weber \((\rm{Wb})\).
Magnetic flux is given by the dot product of the magnetic field and the area vector:

Fig: Direction of the Magnetic Field and the Area.

\(\phi = \overrightarrow B \cdot \overrightarrow S \)
Where,
\(\overrightarrow B\) is the magnetic field.
\(\overrightarrow S\) is the area vector.

Gauss law

Gauss law for magnetic is one of maxwell’s four equations for electromagnetism.
According to this law, the magnetic flux through any closed surface will be zero.
\({\phi _{{\text{closed}}\;{\text{surface}}}} = \oint {\overrightarrow B } \cdot \overrightarrow A = 0\)
Where,
\(\overrightarrow B\) is the magnetic field.
\(\overrightarrow A\) is the area vector.

Magnetic Force

We know that a magnetic field is produced by a magnet or a charge in motion. The magnetic field can apply some force to magnetic materials. Interestingly, the magnetic field cannot apply a force on a stationary charge but can apply force on a moving charge.
The force on a charge in motion is given by the charge multiplied by the cross-product of the velocity and magnetic field.

Fig 4: Charge Moving with a Velocity in a Uniform Magnetic Field.

\(\overrightarrow F = q\left( {\overrightarrow v \times \overrightarrow B } \right)\)
Where,
\(q\) is the value of the charge.
\(\overrightarrow v\) is the velocity of the charge
\(\overrightarrow B\) is the magnetic field.

Biot-Savart Law

Biot-savart law, gives us the magnitude and the direction of the magnetic field produced by a current element.
According to this law,
\({\text{d}}\overrightarrow B \propto \frac{{{\text{d}}\overrightarrow l \times \overrightarrow r }}{{{r^3}}}\)
\({\text{d}}\overrightarrow B = \frac{{{\mu _0}}}{{4\pi }}\frac{{I{\text{d}}\overrightarrow l \times \overrightarrow r }}{{{r^3}}}\)
Where,
\(\frac{{{\mu _0}}}{{4\pi }} = {10^{ – 7}}\;{\text{T}}\;{\text{m}}{{\text{A}}^{ – 1}}\) is the proportionality constant.
\(\mu _0\) is known as the permeability of space.
\({\text{d}}\overrightarrow l\) is the length of the current element under consideration.
\(\overrightarrow r\) is the position vector of the point at which we want to calculate the magnetic field.
\(I\) is the current flowing through the current element.

Calculation of Magnetic Field

The magnetic field is a vector quantity, and the total magnetic field at a point is given by the vector sum of individual magnetic fields at that point.
\(\overrightarrow {{B_{{\text{net}}}}} = \overrightarrow {{B_1}} + \overrightarrow {{B_2}} + \overrightarrow {{B_3}} + …..\).
For continuous bodies,
The net magnetic field is given by integrating the magnetic field due to the differential element of the continuous body.
\(\overrightarrow {{B_{{\text{net}}}}} = \int {\text{d}} \overrightarrow B \)

Magnetic Force on a Current Element

Fig: Current Element in a Magnetic Field.

Magnetic force on a current-carrying wire due to a magnetic field is given by,
\(\overrightarrow F = i\left( {{\text{d}}\overrightarrow l \times \overrightarrow B } \right)\)
Fleming’s Left-hand rule
The direction of the force in the above equation is in the direction of the cross product, \(\left( {{\text{d}}\overrightarrow l \times \overrightarrow B } \right)\). Fleming’s left-hand rule simplifies it. According to fleming’s left-hand rule, if we align the index finger in the direction of the magnetic field and the middle finger in the direction of the current, then our thumb points towards the direction of the force on the current-carrying wire.

Magnetic Field due to a Current-Carrying Wire

Fig: A finite-Length Current-Carrying Wire.

\(\overrightarrow B = \frac{{{\mu _0}i}}{{4\pi d}}\left( {{\text{sin}}\left( {{\alpha _1}} \right) + {\text{sin}}\left( {{\alpha _2}} \right)} \right)\)
Where,
\(i\) is the current flowing through the wire.
\(d\) is the distance of the point at which we want to determine the magnetic field.
\(\alpha _1\) and \(\alpha _2\) are the angle subtended by the ends of the with at the point we want to determine the magnetic field.
The direction of the magnetic field is given by the right-hand thumb rule.

The Force between Two Current-Carrying Conductor

Fig: Two Parallel Current-Carrying Conductors.

The force per unit length of the conductor is given by,
\(\frac{{{\text{d}}F}}{{{\text{d}}l}} = \frac{{{\mu _0}{i_1}{i_2}}}{{2\pi d}}\)
Where,
\(i_1\) and \(i_2\) are the magnitude of the current flowing through the two conductors.
\(d\) is the distance between the two wires
The force is attractive in nature if the current in both the wires is in the same direction and repulsive if the current is in a different direction.

Magnetic Field due to Circular Current-Carrying Loop

Fig: A Current-Carrying Loop.

The magnetic field at the center of the loop is given by,
\(\overrightarrow B = \frac{{{\mu _0}i}}{{2a}}\)
The magnetic field due to a circular loop of radius ‘\(a\)’, carrying current ‘\(i\)’ at a distance ‘\(x\)’ from the center on its axis, is given by,
\(\overrightarrow B = \frac{{{\mu _0}i{a^2}}}{{2{{\left( {{a^2} + {x^2}} \right)}^{\frac{3}{2}}}}}\)
The magnetic field at the centre of the loop is given by,
\(\overrightarrow B = \frac{{{\mu _0}i}}{{2a}}\)
The direction of the magnetic field is given by the right-hand curl rule. If we curl our fingers in the direction of the current in the loop, then the thumb will give the direction of the magnetic field.

Fig: Magnetic Field Lines due to a Current-Carrying Loop.

If the radius is very small as compared to the distance of the point at which the magnetic field is to be calculated, then the expression for magnetic field becomes,
\(\overrightarrow B = \frac{{{\mu _0}i{a^2}}}{{2{x^3}}} = \frac{{{\mu _0}i2\pi {a^2}}}{{4\pi {x^3}}}\)
\( \Rightarrow \overrightarrow B = \frac{{{\mu _0}}}{{4\pi }} \cdot \frac{{2\overrightarrow p }}{{{x^3}}}\)
Where,
\(\overrightarrow p = i\pi {a^2}\) is the magnetic dipole moment of the loop.

Ampere’s Law

Ampere’s law states that the line integral of the magnetic field \(\overrightarrow B\) around any closed loop or curve is equal to \(\mu _0\) times the net current enclosed by the loop.

Fig: Ampere Circuital Law

\(\oint {\overrightarrow B \cdot {\text{d}} \overrightarrow l} = {\mu _0}\left( {{i_1} – {i_2} + {i_3}} \right)\)
Where,
\(\mu _0\) is the permeability of the space.

Solenoid

When a current-carrying conductor is looped several times, then the shape obtained is called a solenoid.

Fig: A Solenoid Formed by Coiling the Wire Around the Iron Nail.

Magnetic field due to a solenoid is given by,

Fig: A solenoid

Fig: Lateral view of the solenoid

\(B = \frac{{{\mu _0}}}{{4\pi }}\left( {2\pi ni} \right)\left[ {\sin \alpha + \sin \beta } \right]\)
Where,
\(n =\) number of turns per unit length
\(i\) is the current passing through the conductor
For an infinite length solenoid
\(\overrightarrow B = {\mu _0}ni\)
Assumptions
1. The field inside the solenoid is uniform.
2. The field outside the solenoid is zero.

Toroid

FIG 14: A Toroid

When the ends of the solenoid are joined to form a ring, the resultant shape is known as a toroid.
The magnetic field inside the toroid is given by,
\(\overrightarrow B = {\mu _0}ni\)
Where, \(n = \frac{N}{{2\pi r}}\)
\(n =\) number of turns per unit length
\(i\) is the current passing through the conductor
\(\mu _0\) is the permeability of the space.
Assumptions
1. The field inside the toroid is uniform.
2. The field outside the toroid is zero.

Solved Examples on Magnetic Effects Of Electric Current

Q.1. If the current flowing through the two wires is \(2\;\rm{A}\), then find the force per unit length between them if the distance between them is \(4\;\rm{m}\).
Ans: Given,
The current flowing in wire \(1\) is \(2\;\rm{A}\).
The current flowing in wire \(2\) is \(2\;\rm{A}\).
The distance between the two wires is \(4\;\rm{m}\)
The force per unit length of the wire is given by,
\(\frac{{{\text{d}}F}}{{{\text{d}}l}} = \frac{{{\mu _0}{i_1}{i_2}}}{{2\pi d}}\)
Where,
\(i_1\) and \(i_2\) are the magnitude of the current flowing through the two conductors.
\(d\) is the distance between the two wires
Putting in the values, we get,
\(F = \frac{{4\pi \times {{10}^{ – 7}} \times 2 \times 2}}{{2\pi \times 4}} = 2 \times {10^{ – 7}}\;{\text{N}}\).

Q.2. A circular loop is carrying a current ‘\(I\)’, as shown in the figure. What happens to the loop by applying a uniform magnetic field inward perpendicular to the plane of the loop? Will it expand or contract?
Ans: The direction of the magnetic force will be given by Biot-savart law,
\({\text{d}}\overrightarrow B = \frac{{{\mu _0}}}{{4\pi }}\frac{{I\;{\text{d}}\overrightarrow l \times \overrightarrow r }}{{{r^3}}}\)
Where,
\(\frac{{{\mu _0}}}{{4\pi }} = {10^{ – 7}}\;{\text{T}}\;{{\text{mA}}^{ – 1}}\)
\(\mu _0\) is known as permeability of space.
\({\text{d}}\overrightarrow l\) is the length of the current element under consideration.
\(\overrightarrow r\) is the position vector of the point at which we want to calculate the magnetic field.
\(I\) is the current flowing through the current element.
Since the magnetic field is inward and the current is clockwise, therefore, the force on the loop is in an outward direction; therefore, the radius of the loop will increase, or we can say that the loop will expand.

Applications of Magnetic Effects of Electric Current

The magnetic effect of current is used in an electromagnet.

Fig: An Electromagnet

The magnetic effect of current is used in motors.

Fig: An electric motor

Summary

From the article, we can infer that magnetism and electric current are closely related. A moving charge produces a magnetic field. Magnets can also produce a magnetic field.
Magnetic force on a current-carrying wire is given by,
\(\overrightarrow F = i\left( {{\text{d}}\overrightarrow l \times \overrightarrow B } \right)\)
While the magnetic field produced by a current-carrying wire is given by Biot-savart law, if two current-carrying wires are placed nearby, then they exert force on each other.
Ampere’s circuital law can be used to determine the magnetic field in many cases. According to ampere’s law, the line integral of the magnetic field \(\overrightarrow B \) around any closed loop or curve is equal to \(\mu _0\) times the net current enclosed by the loop.
A solenoid is several circular loops of conductors stacked in a line. The magnetic field outside the solenoid is zero, and it has a uniform magnetic field inside it.
When the ends of the solenoid are joined to form a ring, it is known as a toroid.

FAQs on Magnetic Effects of Electric Current

Q.1. Can a stationary charge experience a Magnetic force?
Ans: No, a stationary force cannot experience a magnetic force. It needs to have some velocity to experience a magnetic force.

Q.2. What is the magnetic force on the charge moving with some velocity?
Ans: The force on a moving charge is given by,
\(\overrightarrow F = q\left( {\overrightarrow v \times \overrightarrow B } \right)\)
Where,
\(q\) is the value of the charge.
\(\overrightarrow v\) is the velocity of the charge
\(\overrightarrow B\) is the magnetic field.

Q.3. What is Biot-savart law?
Ans: Biot-savart law describes the magnitude and the direction of the magnetic field due to a current element.
\({\text{d}}\overrightarrow B = \frac{{{\mu _0}}}{{4\pi }}\frac{{I\;{\text{d}}\overrightarrow l \times \overrightarrow r }}{{{r^3}}}\)
Where,
\(\frac{{{\mu _0}}}{{4\pi }} = {10^{ – 7}}\;{\text{T}}\;{{\text{mA}}^{ – 1}}\) is the proportionality constant.
\(\mu _0\) is known as the permeability of space.
\({\text{d}}\overrightarrow l \) is the length of the current element under consideration.
\(\overrightarrow r \) is the position vector of the point at which we want to calculate the magnetic field.
\(I\) is the current flowing through the current element.

Q.4. Why do magnetic field lines form a closed-loop?
Ans: The magnetic field line forms a closed-loop because a magnetic monopole does not exist. If the magnetic fields terminate or originate from a point, then that point will be a magnetic monopole which is not possible as we cannot isolate a magnetic monopole; therefore, the magnetic field lines will always form a closed loop.

Q.5. What is the difference between the fleming left-hand rule and Flemings right-hand rule?
Ans: Fleming’s left-hand rule is used to determine the force on a current-carrying conductor in a magnetic field. Fleming’s right-hand rule is used to find the direction of induced current in a conductor moving in a magnetic field.

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