Biot-Savart Law MCQ Quiz - Objective Question with Answer for Biot-Savart Law - Download Free PDF

CONCEPT:

Biot - Savart's law gives the equation of magnetic field produced by the current carrying segment and it is written as;

\(d \vec B = \frac {\mu_o I d \vec I\times\hat r }{4 \pi r^2}\)

Here we have B as the magnetic field, I is the current, and r is the distance.

EXPLANATION:

 Biot-Savart's law gives us the expression for the magnetic field strength of an infinitesimal current element (IdI) of a current carrying conductor only which is written as;

\(d \vec B = \frac {\mu_o I d \vec I\times\hat r }{4 \pi r^2}\)

Therefore Statement I is incorrect.

Statement II -  In statement II is given that Biot-Savart's law is analogous to Coulomb's inverse square law of charge q, with the former being related to the field produced by a scalar source, Idl while the latter being produced by a vector source, q. The Biot- Savart's law is the equation of the magnetic field produced by the current carrying conductor whereas in coulomb's law we find the force of attraction and repulsion between the two masses. Biot-Savart's law is not analogous to Coulomb's inverse square law of charge q.

Therefore, this statement is false.

Hence, option 3) is the correct answer.

Calculation:

\(T = MB \sin \theta \)

\(T = I \pi a^2 \times \dfrac{\mu_0 I}{ \pi d} \sin 30^{\circ} \)

∴ \(T = \dfrac{\mu_0 I^2 a^2}{2d}\)

The net magnetic field at the given point will be zero if.

\(\left| \vec{B}_{wires} \right| = \left| \vec{B}_{loop} \right|\)

\(2{ B }_{ w }\cos { \theta } ={ B }_{ l }\)

\(\Rightarrow 2 \dfrac{\mu_0 I}{2 \pi \sqrt{a^2 + h^2}} \times \dfrac{a}{\sqrt{a^2 + h^2}} = \dfrac{\mu_0 Ia^2}{2 (a^2 +h^2)^{3/2}}\)

\(\Rightarrow h \approx 1.2 a\)

The direction of magnetic field at the given point due to the loop is normally out of the plane. Therefore, the net magnetic field due to the both wires should be into the plane. For this current in wire 1 should be along PQ and that in wire 2 should be along SR.

\(i_1+i_2=i\)

In loop ABCD

\(i_2r=i_12r\)

\(\Rightarrow i_2=2i_1\)

\(\Rightarrow i_1+2i_1=i\Rightarrow i_1=\dfrac{i}{3}\)

\(\Rightarrow i_2=\dfrac{2i}{3}\)

Magnetic field due to wire

\(B=\dfrac{\mu_0i}{4\pi d}(\cos\theta_1-\cos\theta_2)\)

\(B_{AB}=\dfrac{\mu_0i_2}{4\pi\dfrac{a}{2}}(\cos 45^0-\cos 135^0)\)

\(=\dfrac{\mu_0i_2}{2\pi a}(\cos 45^0+\sin 45^0)\)

\(=\left(\dfrac{\mu_0 2i}{2\pi a.3}\right)\left(\dfrac{2}{\sqrt{2}}\right)\)

\(B_{ABC}=\dfrac{4\mu_0i}{3\sqrt{2}\pi a}\) (inside the plane of paper)

\(B_{ADC}=\dfrac{2\mu_0i_1}{2\pi a}\left(\dfrac{2}{\sqrt{2}}\right)=\dfrac{2\mu_0i}{3\sqrt{2}\pi a}\) (outside the plane of paper)

\(B_{net}=B_{ABC}-B_{ADC}=\dfrac{2\mu_0i}{3\sqrt{2}\pi a}\)

\(\therefore \boxed{B_{net}=\dfrac{\sqrt{2}\mu_0i(\sqrt{2})}{3\sqrt{2}\pi a}}\Rightarrow \boxed{B_{net}=\dfrac{\sqrt{2}\mu_0i}{3\pi a}}\)

We have to consider the magnetic vector field \(\vec {B}\) at \((x, y)\) in the \(z = 0\) plane.

Magnetic field \(\vec {B}\) is perpendicular to \(OP\).

\(\therefore \vec {B} = B\sin \theta \hat {i} - B\cos \theta \hat {j}\)

\(\sin \theta = \dfrac {y}{r}, \cos \theta = \dfrac {x}{r} B = \dfrac {\mu_{0}I}{2\pi r}\)

\(\therefore \vec {B} = \dfrac {\mu_{0}I}{2\pi r^{2}} (y\hat {i} - x\hat {j})\)

or \(\vec {B} = \dfrac {\mu_{0}I (y\hat {i} - x\hat {j})}{2\pi (x^{2} + y^{2})}\).

CONCEPT:

• Biot-Savarts Law: The law that gives the magnetic field generated by a constant electric current is Biot-savarts law.
• Let us take a current-carrying wire of current I and we need to find the magnetic field at a distance r from the wire then it is given by:

\(dB = \;\frac{{{\mu _0}\;I}}{{4\pi }}\left( {\frac{{\overrightarrow {dl} \times \hat r}}{{{r^2}}}} \right)\)

Where μ0= 4π × 10-7 T.m/A is the permeability of free space/vacuum, dl = small element of wire and r ̂ is the unit position vector of the point where we need to find the magnetic field.

EXPLANATION:

From the above expression of the Biot-savart law, the magnetic field is: 

• Directly proportional to the length of the wire. So statement (i) is correct.
• Directly proportional to the electric current. So statement (iii) is correct.
• Biot-savart law gives the magnetic field, not the electric field. So statement (ii) is wrong.

CONCEPT:

• Biot-Savart Law: The law that gives the magnetic field generated by a constant electric current is Biot-savart law.
• Let us take a current-carrying wire of current I and we need to find the magnetic field at a distance r from the wire then it is given by:

\(dB = \;\frac{{{\mu _0}\;I}}{{4\pi }}\left( {\frac{{\overrightarrow {dl} \times \hat r}}{{{r^2}}}} \right)\)

Where μ0= 4π × 10-7 T.m/A is the permeability of free space/vacuum, dl = small element of wire and r ̂ is the unit position vector of the point where we need to find the magnetic field.

EXPLANATION:

1. Ampere's law also gives the magnetic field but it is not a valid for all the current-carrying wire. There are many restrictions on this law.
2. Lorentz's law gives the force on a moving charge particle in a magnetic and electric field.
3. The Biot-Savart's Law is an equation that describes the magnetic field created by a current-carrying wire and allows you to calculate its strength at various points. So option 3 is correct.
4. Kirchhoff’s law is used to calculate the electric current in an electric circuit.

CONCEPT:

• Biot-Savart's Law: Biot-Savart’s law is used to determine the magnetic field at any point due to a current-carrying conductor.
• This law is although for infinitesimally small conductor yet it can be used for long conductors.

EXPLANATION:

According to Biot-Savart Law, the magnitude of the magnetic field at point ‘ P ’ is

1. directly proportional to the current through the conductor i.e. dB ∝ I
2. directly proportional to the length of the current element i.e. dB ∝ dl
3. directly proportional to sinθ
4. inversely proportional to the square of the distance from the current element i.e. dB ∝ 1/r²

Combining all these four factors, we get

\(dB \propto \frac{{idl\sin \theta }}{{{r^2}}}\)

Or,

\(dB = k\frac{{idl\sin \theta }}{{{r^2}}}\)

Where the proportionality constant K depends on the medium between the observation point P and the current element and the system chosen. For free space and SI units,

\(dB = \frac{{{\mu _o}}}{{4\pi }}\frac{{idl\sin \theta }}{{{r^2}}}\)

Where μ_o = Absolute permeability of air or vacuum, \(i\overrightarrow {dl} \) = Current element and r = distance.

CONCEPT:

• Ampere’s Law: Line integral of the magnetic field B around any closed curve is equal to μ0 times the net current I threading through the area enclosed by the curve i.e.
  \(\oint \vec B \cdot \overrightarrow {dl} = {\mu _o}I\)

Where B = magnetic field, μ0 = permittivity of free space and I = current passing through the coil

EXPLANATION:

• The intensity of the magnetic field due to wire of infinite length at a distance r from it is

\(B = \frac{{{\mu _o}}}{{4\pi }}\frac{{2I}}{d}\)

Where μ0 = permittivity of free space, I = current in a wire, d = distance

• From above equation it is clear that the magnetic field at point due to conductor is inversely proportional to the distance i.e., \(B\;\alpha \frac{1}{r}\)
• From above equation it is clear that the magnetic field (B) increases with the increase in current (I) and decreases as the point moves away from the conductor. 

Concept:

Biot Savart Law:

• Biot-savart’s law gives the magnetic field produced due to the current carrying segment.
• This segment is taken as a vector quantity known as the current element.

• The magnitude of the magnetic field dB at a distance r from a current-carrying element dl is found to be proportional to I and the length dl.
• Formula, \(B=\frac{\mu_0I}{4\pi}\frac{d\vec l× \vec r}{|\vec r|^3}\)
• Here, \(\frac{\mu_0}{4\pi}= 10^{-7} Tm/A\)
• The magnetic field at the point r from the straight wire is, \(B=\frac{\mu_0}{4\pi}\frac{2I}{r}\)

Calculation:

Given,

Current, I = 20 A

Distance from the wire, r = 10 cm

The magnetic field at the point r from the wire is,

\(B=\frac{\mu_0}{4\pi}\frac{2I}{r}\)

\(B=10^{-7}\times \frac{2\times 20}{10\times 10^{-2}}=4\times 10^{-5}Wb/m^2\)

Hence, the magnetic field is 4 × 10-5 Wb/m².

CONCEPT:

Magnetic field lines: The imaginary lines which represent the direction of the magnetic field are called as magnetic field lines.

• The tangent at any point on the field lines gives the direction of the magnetic field vector at that point.
• Magnetic lines of force always emerge or start from the North Pole and terminate on the South Pole.

EXPLANATION:

• A magnetic monopoles (either North Pole or South Pole) never exists alone due to which the magnetic field lines cannot emanate from a point or terminate at a point.
• The two magnetic field lines do not intersect each other because if they do it means at the point of intersect the compass needle is showing two different directions which is not possible.
• There can't be such straight field lines between magnetic poles as the magnetic field always form closed loops. 
• There are continuous field lines exists inside the bar magnet. 

Concept-

Biot-Savart Law:

• The law who gives the magnetic field generated by a constant electric current is Biot-savart law.
• Let us take a current carrying wire of current I and we need to find the magnetic field at a distance r from the wire then it is given by:

\(dB = \;\frac{{{\mu _0}\;I}}{{4\pi }}\left( {\frac{{\overrightarrow {dl} \times \hat r}}{{{r^2}}}} \right)\) = magnetic field due to current carrying wire element dl at the point

\(d\vec B = \;\frac{{{\mu _0}\;I}}{{4\pi }}\left( {\frac{{\overrightarrow {dl} \times \vec r}}{{{r^3}}}} \right)\)

Where,

μ₀ = the permeability of free space/vacuum (4π × 10-7 T.m/A),

dl = small element of wire

\(\hat{r}\)= the unit position vector of the point where we need to find the magnetic field.

Explanation-

\(d\vec B = \frac{{{\mu _0}\;I}}{{4\pi }}\left( {\frac{{\overrightarrow {dl} \times \vec r}}{{{r^3}}}} \right)\)

Concept:

Ampere's law : 

It states that the magnetic field created by an electric current is proportional to the size of that electric current with a constant of proportionality equal to the permeability of free space.

The magnetic field due to long  straight wire is given by;

\({\rm{B}} = \frac{{{{\rm{\mu }}_0}{\rm{I}}}}{{2{\rm{\pi R}}}}\)

Where B = strength of magnetic field, I = current, R = radius or distance.

Explanation:

Given:

R₁ = 4 cm = 0.04 m, R₂ = 2 cm = 0.02 m, B₁ = 10^-3

So as from the Amperes law

\( \frac{{{B_1}}}{{{B_2}}} = \frac{{{R_2}}}{{{R_1}}}\)

\( {B_2} = {B_1}\frac{{{R_1}}}{{{R_2}}} \)

\( {B_2} = {10^{ - 3}}\times \frac{4}{{2}} = 2\times {10^{ - 3}}\;T \)

CONCEPT:

• Ampere’s Law: Line integral of the magnetic field B around any closed curve is equal to μ0 times the net current I threading through the area enclosed by the curve i.e.
  \(\oint \vec B \cdot \overrightarrow {dl} = {\mu _o}I\)

Where B = magnetic field, μ0 = permittivity of free space and I = current passing through the coil

EXPLANATION:

Given - B₁ = 0.2 T and r₁ = 2

• The intensity of the magnetic field due to wire of infinite length at a distance r from it is

\(B = \frac{{{\mu _o}}}{{4\pi }}\frac{{2I}}{d}\)

Where μ0 = permittivity of free space, I = current in a wire, d = distance

• As current is constant in the wire, then the magnetic field varies with the distance r as

\(\Rightarrow B\;\alpha \frac{1}{r}\)

\(\Rightarrow B_1r_1=B_2r_2\)

\(\Rightarrow B_2=\frac{B_1r_1}{r_2}\)

When the distance is doubled (r₂ = 2r), then the magnetic field is

\(\Rightarrow B_2=\frac{0.4\times r}{2r}=0.2T\)

• Thus, the magnetic field changes to 0.2T at a distance 2r from the wire.

Concept:

The magnetic field at the center of the circular coil:

• The circular loop can be supposed to consist of small elements placed side by side.
• The magnetic field due to all elements will be in the same direction.
• 
• The magnetic field at the center P, \(B=\frac{μ_0 I}{2r}\)
• For n number of coils, \(B=\frac{μ_0n I}{2r}\)

Calculation:

Given, Current, I = 1A, radius, r = 1m

The magnetic field at the center of the circular loop, 

\(B=\frac{μ_0 I}{2r}\)

\(B=\frac{μ_0 \times 1}{2\times 1}=\frac{μ_0}{2}\)

Hence, the magnetic field at the center of the circular loop is μ₀/2. 

Additional Information

Biot Savart Law:

• Biot-savart’s law gives the magnetic field produced due to the current carrying segment.
• This segment is taken as a vector quantity known as the current element.

• The magnitude of the magnetic field dB at a distance r from a current-carrying element dl is found to be proportional to I and the length dl.
• Formula, \(B=\frac{\mu_0I}{4\pi}\frac{d\vec l× \vec r}{|\vec r|^3}\)
• Here, \(\frac{\mu_0}{4\pi}= 10^{-7} Tm/A\)