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

$d\overrightarrow{B}=\frac{{\mu }_{0}I}{4\pi }\left(\frac{\overrightarrow{dl}\times \overrightarrow{r}}{r^3}\right)$

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-

$d\overrightarrow{B}=\frac{{\mu }_{0}I}{4\pi }\left(\frac{\overrightarrow{dl}\times \overrightarrow{r}}{r^3}\right)$

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.

$\oint \overrightarrow{B}\cdot \overrightarrow{dl}={\mu }_{o}I$

• The intensity of the magnetic field due to wire of infinite length is​

$B=\frac{{\mu }_o}{2\pi }\frac{I}{d}$

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

CALCULATION:

Given - Magnitude of current (I) = 1 A and distance between wire and at some point (r) = 2 m

• The intensity of the magnetic field due to wire of infinite length is​

$\Rightarrow B=\frac{{\mu }_o}{2\pi }\frac{I}{d}$

$\Rightarrow B=\frac{{\mu }_{0}I}{2\pi d}=\frac{4\pi \times {10}^{-7}\times 1}{2\times \pi \times 2}={10}^{-7}Wb$

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.

$\oint \overrightarrow{B}\cdot \overrightarrow{dl}={\mu }_{o}I$

where I - current flowing

EXPLANATION:

• In a simplified form, Ampere's circuital law states that if the field  is directed along the tangent to every point on the perimeter L of a closed curve and its magnitude is constant along the curve, then BL = μ_oI

where I = the net current enclosed by the closed circuit.

• The closed curve is called the Amperean loop which is a geometrical entity and not a real wire loop.

• According to this law, magnetic fields are related to the electric current produced in them.
• The law specifies the magnetic field that is associated with a given current or vice-versa, provided that the electric field doesn’t change with time.
• It can be applied outside the conductor only. Therefore option 2 is correct.

CONCEPT:

Amperes law: The line integral of the magnetic field around any closed curve is equal to μo times the net current I threading through the area enclosed by the curve.

$\oint \overrightarrow{B}\cdot \overrightarrow{dl}={\mu }_{0}I$

Ampere’s law in this form is not valid if the electric field at the surface varies with time.

Here,

B = magnetic field intensity

I = Electric current 

dl = small length

μ₀ =permeability of free space

Explanation:

From the above explanation, we can see that

• The “Ampère's circuital law” draws a relationship between the integrated magnetic field around a closed loop and the electric current passing through the loop.
• According to the law, the product of electric current passing through the closed-loop and the permeability of the medium is equal to the integral of magnetic field density (B) along an imaginary closed path.

Hence only option 2 is correct among all

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.
  $\oint \overrightarrow{B}\cdot \overrightarrow{dl}={\mu }_{o}I$

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

$\Rightarrow B\alpha \frac{1}{r}$

$\Rightarrow B_1{r}_1=B_2{r}_2$

$\Rightarrow B_2=\frac{B_1{r}_1}{r_2}$

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.

The correct answer is Oersted.

CONCEPT:

• Oersted’s experiment: Hans Christian Oersted was a Danish scientist who explored the relationship between electric current and magnetism.
  • The wire will carry a current that creates a magnetic field around itself. Bringing the compass near the wire or in the loop will cause the compass needle to move.
  • The current had produced a magnetic field strong enough to cause the compass needle to turn.

EXPLANATION:

• Oersted’s experiment explains that when an electric current passes through a conducting wire, a magnetic field is produced around the wire. 
• Thus the magnetic effect of current was discovered by Oersted. So option 3 is correct.

B-H relation:

The relation between magnetic field intensity (H) and magnetic flux density (B) is

B = μ H Wb/m2

μ = The magnetic permeability

And $H=\frac{NI}{l}$

$\Rightarrow B=\frac{\mu NI}{l}$

CONCEPT:

• Magnetic field and magnetic lines of force: It is the space around a magnetic pole or magnet or current-carrying wire within which its magnetic effect can be experienced is defined as a magnetic field.

• The magnetic field can be represented with the help of a set of lines or curves called magnetic lines of force or magnetic field lines.

Properties of magnetic field line:

1. The magnetic field line is directed from the north pole to the south pole outside and south to the north inside the magnet.
2. Magnetic field lines are closed and continuous.
3. Magnetic field lines are more crowded near poles which shows that the strength of the magnetic field is maximum at its poles.
4. Magnetic field lines never intersect with each other.

EXPLANATION:

From the above explanation,

• We can see that magnetic field lines originate from the north pole and terminate at the south pole while forming continuous closed paths. 
• But they don't start or end at any point, they are continuous and follows a closed-loop

Hence option 2 is the only incorrect among the following.

1 tesla = 104 gauss

Explanation:

SI unit of the magnetic field (B) is weber/meter² (Wbm^-2) or tesla.

The CGS unit of magnetic field (B) is gauss where 1 gauss = 10-4 tesla.

From the above explanation, we can see that the relation between tesla and gauss is 

1 tesla = 10⁴ gaussKey Points

Magnetic field strength:

Magnetic field strength or magnetic field induction or flux density of the magnetic field is equal to the force experienced by a unit positive charge moving with unit velocity in a direction perpendicular to the magnetic field.

The correct answer is option 4) i.e. 

CONCEPT:

• Magnetic field due to a straight current-carrying conductor: Biot-Savart Law
  • ​Magnetic field B at a radial distance r, due to a wire carrying current is given by:

$B=\frac{{\mu }_{0}I}{2\pi r}$

Where μ0 is the permeability of free space (4π × 10-7 Tm/A), and I is the current intensity.

EXPLANATION:

Magnetic field intensity, $B=\frac{{\mu }_{0}I}{2\pi r}$ 

$\Rightarrow B\propto \frac{1}{r}$

Thus, the magnetic field is inversely proportional to the distance from the current-carrying wire.

This is represented by the graph in option 4).