Question

If you want to construct an electromagnet by running a current of 3.00 A through a solenoid with 500 . windings and length \(3.50 \mathrm{~cm}\) and you want the magnetic field inside the solenoid to have the magnitude \(B=2.96 \mathrm{~T}\) you can insert a ferrite core into the solenoid. What value of the relative magnetic permeability should this ferrite core have in order to make this work?

Short answer

Answer: The relative magnetic permeability of the ferrite core required is approximately 55.21.

Step-by-step solution

Calculate the Magnetomotive Force (MMF)

The magnetomotive force (MMF) of a solenoid is given by the formula: \[\text{MMF} = NI\] where N is the number of windings and I is the current passing through the solenoid. Given N = 500 windings and I = 3.00 A, we can calculate the MMF: \[\text{MMF} = (500)(3.00\,\text{A}) = 1500\,\text{A}\text{-}\text{turns}\]

Calculate the Magnetic Field in Air (B_air)

To calculate the magnetic field in air (B_air) generated by the solenoid, we need to use Ampere's Law, which is given by the formula: \[B_air = \frac{\mu_0 \times \text{MMF}}{L}\] where \(\mu_0 \approx 4\pi\times 10^{-7}\,\text{T}\cdot\text{m}\cdot\text{A}^{-1}\) is the vacuum permeability (magnetic constant), MMF is the magnetomotive force calculated in Step 1, and L is the length of the solenoid. Given L = 3.50 cm = 0.035 m, we can calculate B_air: \[B_air = \frac{4\pi\times 10^{-7}\,\text{T}\cdot\text{m}\cdot\text{A}^{-1} \times 1500\,\text{A}\text{-}\text{turns}}{0.035\,\text{m}} \approx 0.05363\,\text{T}\]

Calculate the Relative Magnetic Permeability (µ_r)

The relative magnetic permeability (µ_r) can be determined by comparing the desired magnetic field (B) with the magnetic field in air (B_air). Since the relationship between the magnetic field in the core and air (B_ferrite and B_air, respectively) and their corresponding magnetic permeabilities is given by B_ferrite = µ_r × B_air, we can find µ_r by rearranging this equation: \[\mu_r = \frac{B_\text{ferrite}}{B_\text{air}}\] Given B_ferrite = 2.96 T and B_air ≈ 0.05363 T, we can calculate µ_r: \[\mu_r = \frac{2.96\,\text{T}}{0.05363\,\text{T}} \approx 55.21\] The ferrite core should have a relative magnetic permeability of approximately 55.21 in order to make the electromagnet work as desired.

Key concepts

Magnetomotive Force

Magnetomotive force (MMF) is like the 'pressure' that drives the magnetic field through a circuit, similar to how voltage pushes electrical current through a wire. It's measured in units of ampere-turns (A-t), which reflects the product of the current (in amperes) and the number of turns or coils in the wire. For an electromagnet, which is a type of magnetic circuit, the MMF plays a crucial role in determining the strength of the magnetic field it can produce.

To calculate the MMF, you can use the simple formula \[ \text{MMF} = NI \], wherein N is the total number of windings or turns in the coil, and I is the current passing through the coil. In the specified exercise, with 500 windings and a current of 3.00 A, the MMF is calculated as \[ 1500 \, \text{A-turns} \]. This value is an indicator of the magnet's potential to generate a magnetic field before considering the magnetic properties of the core material.

Ampere's Law

Ampere's Law, named after André-Marie Ampère, relates magnetic fields to the electric currents that produce them. It's a foundational principle in electromagnetism, stating that for any closed loop path, the sum of the length elements times the magnetic field in the direction of the length element is equal to the permeability times the electric current enclosed in the loop.

For a straight solenoid, Ampere's Law simplifies to the formula \[ B = \frac{\mu_0 \times \text{MMF}}{L} \], where B is the magnetic field, \(\mu_0\) is the magnetic constant (also known as the permeability of free space), MMF is the magnetomotive force, and L is the length of the solenoid. This equation allows us to calculate the theoretical strength of the magnetic field generated in the air, or vacuum, inside the solenoid without any magnetic material present.

Relative Magnetic Permeability

Relative magnetic permeability (\(\mu_r\)) is a dimensionless value that compares a material's ability to conduct magnetic flux relative to the vacuum of space. The value tells us how much more or less effective the material is at magnetizing compared to empty space, which has a \(\mu_r\) of 1. Most materials have a relative magnetic permeability greater than 1, making them better at channeling magnetic field lines than a vacuum.

When creating an electromagnet, by choosing a material with a high \(\mu_r\), we enhance the magnetic field's strength inside the electromagnet. This property is vital when selecting core materials for inductors, transformers, or electromagnets. Once you know the desired magnetic field and the magnetic field in the air, you can compute the required relative permeability using the ratio \[ \mu_r = \frac{B_{\text{desired}}}{B_{\text{air}}} \]. In our exercise, to achieve a magnetic field of 2.96 T inside the ferrite core, the calculated \(\mu_r\) of approximately 55.21 indicates how many times the ferrite core must be better than a vacuum at magnetizing.

Solenoid Magnetic Field

The magnetic field within a solenoid is unique because it's fairly uniform and parallel to the axis of the solenoid, providing an almost constant field strength. This characteristic makes solenoids useful for creating controlled magnetic fields. To gain the magnetic field inside the solenoid, we begin by calculating the field in air, assuming no core material is present, and then adjusting this value based on the material's permeability that we insert into the solenoid.

The formula to find the magnetic field of a solenoid in air is \[ B_{\text{air}} = \frac{\mu_0 \times \text{MMF}}{L} \] as derived from Ampere's Law. But with a material inside, the field is enhanced and given by \[ B_{\text{ferrite}} = \mu_r \times B_{\text{air}} \]. In our exercise example, with a known current and solenoid dimensions, we required a specific \(\mu_r\) to reach the desired 2.96 T magnetic field within the solenoid, which demonstrates the direct influence of the core material's properties on the electromagnet's performance.