understand that the magnetic field due to the current in a solenoid is increased by a ferrous core

Magnetic Fields Due to Currents

Objective

Understand that the magnetic field inside a solenoid is increased when a ferrous core is inserted, and be able to calculate the resulting field.

1. Magnetic Field of a Long Solenoid

A long solenoid of length \(l\), carrying a current \(I\) through \(N\) turns, produces a magnetic field approximately uniform inside:

• Number of turns per unit length: \(n = \dfrac{N}{l}\)
• Field without core: \(B = \mu_0 n I\)

where \(\mu_0 = 4\pi\times10^{-7}\,\text{H/m}\) is the permeability of free space.

2. Effect of a Ferrous Core

When a core of relative permeability \(\mu_r\) is inserted, the magnetic field becomes

\[ B_{\text{core}} = \mu_0 \mu_r n I \]

The core increases the field by a factor of \(\mu_r\). Typical ferrous materials have \(\mu_r\) ranging from 100 to 10,000.

3. Magnetic Permeability

The absolute permeability of the core is

\[ \mu = \mu_0 \mu_r \]

For a linear material, \(\mu_r\) is constant. In practice, \(\mu_r\) decreases with increasing magnetic field strength due to saturation.

4. Example Calculation

Calculate the magnetic field inside a solenoid of length \(0.5\,\text{m}\), with \(2000\) turns, carrying a current of \(2\,\text{A}\). The solenoid is filled with a core of relative permeability \(\mu_r = 500\).

1. Compute \(n = \dfrac{2000}{0.5} = 4000\,\text{turns/m}\).
2. Compute \(B_{\text{core}} = \mu_0 \mu_r n I = (4\pi\times10^{-7})(500)(4000)(2)\).
3. Evaluate: \(B_{\text{core}} \approx 0.0050\,\text{T}\) (5 mT).
4. For comparison, without the core: \(B = \mu_0 n I \approx 0.00001\,\text{T}\) (10 µT).

Thus the core increases the field by a factor of 500.

5. Summary Table

6. Practice Questions

1. For a solenoid with \(N = 1500\) turns, \(l = 0.3\,\text{m}\), and \(I = 3\,\text{A}\), calculate the magnetic field inside the solenoid when:
   1. It has an air core.
   2. It contains a core with \(\mu_r = 200\).
2. Explain qualitatively why the magnetic field increases when a ferrous core is inserted.
3. Describe what happens to the magnetic field as the core material approaches magnetic saturation.

7. Suggested Diagram