The space inside a straight current carrying solenoid is filled with a magnetic material having magnetic susceptibility equal to 1.2 × 10–5. What is fractional increase in the magnetic field inside solenoid with respect to air as medium inside the solenoid?
1.2 × 10–5
1.2 × 10–3
1.8 × 10–3
2.4 × 10–5
The Correct Option is A
Approach Solution - 1
To find the fractional increase in the magnetic field inside a solenoid when filled with a magnetic material, we need to understand the effect of magnetic susceptibility on the magnetic field inside the solenoid.
1. **Magnetic Field Inside a Solenoid:**
For a solenoid with air inside, the magnetic field \(B_0\) is given by:
\(B_0 = \mu_0 \cdot n \cdot I\)
where:
- \(\mu_0\) = permeability of free space
- \(n\) = number of turns per unit length of the solenoid
- \(I\) = current through the solenoid
2. **Effect of Magnetic Material:**
When a magnetic material with susceptibility \(\chi_m\) is inserted inside the solenoid, the permeability becomes:
\(\mu = \mu_0 (1 + \chi_m)\)
The new magnetic field \(B\) inside the solenoid is:
\(B = \mu \cdot n \cdot I = \mu_0 (1 + \chi_m) \cdot n \cdot I\)
3. **Fractional Increase in the Magnetic Field:**
The fractional increase in the magnetic field is given by:
\(\text{Fractional increase} = \frac{B - B_0}{B_0} = \frac{\mu_0 (1 + \chi_m) - \mu_0}{\mu_0} = \chi_m\)
Given \(\chi_m = 1.2 \times 10^{-5}\), the fractional increase in the magnetic field is simply this susceptibility value:
\(\text{Fractional increase} = 1.2 \times 10^{-5}\)
Therefore, the correct answer is the fractional increase is \(1.2 \times 10^{-5}\).
Approach Solution -2
The correct answer is (A) : 1.2 × 10–5
\(\stackrel{→}{B^′}=μ_0(1+X)ni\) in the material
\(\stackrel{→}{B}=μ_0ni\) without material
So fractional increase is
\(\frac{B^′−B}{B}=X=1.2×10^{−5}\)’
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Concepts Used:
Magnetic Field
The magnetic field is a field created by moving electric charges. It is a force field that exerts a force on materials such as iron when they are placed in its vicinity. Magnetic fields do not require a medium to propagate; they can even propagate in a vacuum. Magnetic field also referred to as a vector field, describes the magnetic influence on moving electric charges, magnetic materials, and electric currents.
A magnetic field can be presented in two ways.
- Magnetic Field Vector: The magnetic field is described mathematically as a vector field. This vector field can be plotted directly as a set of many vectors drawn on a grid. Each vector points in the direction that a compass would point and has length dependent on the strength of the magnetic force.
- Magnetic Field Lines: An alternative way to represent the information contained within a vector field is with the use of field lines. Here we dispense with the grid pattern and connect the vectors with smooth lines.
Properties of Magnetic Field Lines
- Magnetic field lines never cross each other
- The density of the field lines indicates the strength of the field
- Magnetic field lines always make closed-loops
- Magnetic field lines always emerge or start from the north pole and terminate at the south pole.