Question

If you want to construct an electromagnet by running a current of 3.0 A through a solenoid with 500 windings and length \(3.5 \mathrm{~cm}\) and you want the magnetic field inside the solenoid to have 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 needed is approximately 699.43.

Step-by-step solution

Known values and constants

We are given the following information: - Current, \(I = 3.0 \mathrm{~A}\) - Number of windings, \(N = 500\) - Solenoid length, \(l = 3.5 \mathrm{~cm} = 0.035 \mathrm{~m}\) - Magnetic field, \(B = 2.96 \mathrm{~T}\) - Permeability of free space, \(\mu_0 = 4\pi \times 10^{-7} \mathrm{~Tm/A}\)

Calculate the number of turns per unit length

We can calculate the number of turns per unit length (\(n\)) by dividing the total number of windings by the length of the solenoid: \(n = \frac{N}{l} = \frac{500}{0.035 \mathrm{~m}} = 14285.71 \mathrm{~turns/m}\)

Rearrange the formula for the magnetic field

We rearrange the formula for the magnetic field inside a solenoid with a core to solve for \(\mu_r\): \(\mu_r = \frac{B}{\mu_0 n I}\)

Plug in the values and solve for the relative magnetic permeability

Now we can plug in the values and solve for \(\mu_r\): \(\mu_r = \frac{2.96 \mathrm{~T}}{(4\pi \times 10^{-7} \mathrm{~Tm/A})(14285.71 \mathrm{~turns/m})(3.0 \mathrm{~A})} = 699.43\) The relative magnetic permeability of the ferrite core should be approximately 699.43 to create the desired magnetic field.

Key concepts

Magnetic Field

A magnetic field is an invisible force that surrounds magnetic objects and electric currents. It's crucial to many technologies, such as electromagnets and generators. The strength and direction of a magnetic field are represented by magnetic field lines, which show how a magnetic force would act on a moving charged particle.

Key characteristics of a magnetic field include:

• Direction: From the north pole to the south pole of a magnet.
• Strength: Measured in teslas (T) or gauss, indicating how intense the field is.
• Origin: Produced by moving electric charges, such as in a current-carrying wire.

Understanding these aspects helps in designing effective electromagnetic devices like solenoids, which use magnetic fields generated by currents to perform work.

Solenoid

A solenoid is a long coil of wire wound in a helical pattern. It generates a magnetic field when an electric current passes through it. This field is intense and uniform inside the coil, making solenoids very useful in various applications, such as creating electromagnets and actuators.

Important aspects of solenoids include:

• Windings: The number of turns affects the magnetic field strength, with more turns leading to a stronger field.
• Current: The strength of the magnetic field is directly proportional to the current passing through the coil.
• Core material: Inserting magnetic materials like ferrite can enhance the field due to their high magnetic permeability.

Understanding these factors allows the optimization of solenoids for specific uses, such as those requiring strong and controllable magnetic fields.

Magnetic Permeability

Magnetic permeability is a measure of how easily a material can support the formation of a magnetic field within itself. It's a critical property that affects how materials respond to external magnetic fields. Higher magnetic permeability allows materials to become magnetized more readily, enhancing the magnetic field in devices like solenoids.

Key points about magnetic permeability:

• Relative permeability (\(\mu_r\)): It's a dimensionless number indicating how a material's permeability compares to the permeability of free space (\(\mu_0\)).
• Impact on electromagnets: Materials with high relative permeability, like ferrite, are used to intensify the magnetic field inside a solenoid.
• Relation to core materials: Choosing the right core material can significantly boost a solenoid's magnetic performance.

By selecting materials with appropriate magnetic permeability, engineers and scientists can design efficient magnetic devices crucial for various technologies.