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Magnetic Field due to a Current — Physics STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 SciencePhysicsMagnetic Field due to a Current1 Mark
MCQ
Two parallel, long wires carry currents $i_1$ and $i_2$ with $i_1 > i_2$. When the currents are in the same direction, the magnetic field at a point midway between the wires is $10 \mu\text{T}.$ If the direction of $i_2$ is reversed, the field becomes $30 \mu\text{T}.$ The ratio $\frac{\text{i}_1}{\text{i}_2}$ is :
  • A
    $4$
  • B
    $3$
  • ✓
    $2$
  • D
    $1$
✓

Answer

Correct option: C.
$2$
The magnetic field due to the current$-$carrying long, straight wire at point a is given by,
$\text{B}=\frac{\mu_0\text{i}}{2\pi\text{d}}$
When both the wires carry currents $i_1$ and $i_2$ ​in the same direction, they produce magnetic fields in opposite directions at any point in between the wires.
$\text{B}'=\frac{\mu_0\text{i}_1}{2\pi\text{a}}-\frac{\mu_0\text{i}_2}{2\pi\text{a}}=10\mu\text{T}\ ...(1)$
Here, a is the distance of the midpoint from both the wires.
When both the wires carry currents in opposite directions, they produce fields in the same direction at the midpoint of the two wires.
$\text{B}\ ''=\frac{\mu_0\text{i}_1}{2\pi\text{a}}+\frac{\mu_0\text{i}_2}{2\pi\text{a}}=30\mu\text{T}\ ...(2)$
On solving eqs. $(1)$ and $(2),$ we get
$\text{i}_1-\text{i}_2=10$
$\text{i}_1+\text{i}_2=30$
$\Rightarrow\text{i}_1=20$
$ \text{i}_2=10$
$\frac{\text{i}_1}{\text{i}_2}=\frac{20}{10}$
$\frac{\text{i}_1}{\text{1}_2}=\frac{2}{1}$
$\Rightarrow\frac{\text{i}_1}{\text{i}_2}=2$

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