Magnetic Force Density on an Iron Face: A Detailed Solution

This problem asks us to determine the force density acting on an iron face, given its magnetic flux density. Understanding the relationship between magnetic fields and the forces they exert on materials is crucial in electromagnetism.

Understanding Magnetic Flux Density and Force Density

Magnetic flux density, often denoted by \(B\), is a measure of the strength of a magnetic field. It tells us how much magnetic flux passes through a unit area perpendicular to the direction of the flux. Its unit is Tesla (T).

Force density, in the context of magnetic fields, refers to the force exerted per unit volume or, as in this case, the force exerted per unit area on a surface in a magnetic field. It's often related to magnetic pressure. The unit for force density on a surface is Newtons per square meter (\(\text{N/m}^2\)).

Key Concepts and Formula

When a magnetic field acts on the surface of a ferromagnetic material, like iron, it exerts a pressure or force density. This force density is given by the formula:

\[ f = \frac{B^2}{2\mu_0} \]

Where:

• \(f\) is the force density (\(\text{N/m}^2\)).
• \(B\) is the magnetic flux density (T).
• \(\mu_0\) is the permeability of free space, a fundamental physical constant. Its value is approximately \(4\pi \times 10^{-7}\) H/m (Henry per meter).

Given Data and Calculation Steps

Let's list the given values from the question:

Now, we can substitute these values into the force density formula:

Step 1: Write down the formula for force density.

\[ f = \frac{B^2}{2\mu_0} \]

Step 2: Substitute the given values into the formula.

\[ f = \frac{(1.5 \, \text{T})^2}{2 \times (4\pi \times 10^{-7} \, \text{H/m})} \]

Step 3: Calculate the square of the magnetic flux density.

\[ (1.5)^2 = 2.25 \]

Step 4: Calculate the denominator.

\[ 2 \times 4\pi \times 10^{-7} = 8\pi \times 10^{-7} \]

Using \(\pi \approx 3.14159\):

\[ 8 \times 3.14159 \times 10^{-7} \approx 25.13272 \times 10^{-7} \]

Step 5: Perform the division to find the force density.

\[ f = \frac{2.25}{25.13272 \times 10^{-7}} \]

\[ f \approx 0.08952 \times 10^7 \]

Step 6: Express the result in scientific notation with the correct prefix.

\[ f \approx 0.8952 \times 10^6 \, \text{N/m}^2 \]

Rounding to two decimal places, we get:

\[ f \approx 0.89 \times 10^6 \, \text{N/m}^2 \]

Comparing with Options

Let's compare our calculated force density with the given options:

• Option 1: \(0.59 \times 10^6 \, \text{N/m}^2\)
• Option 2: \(0.89 \times 10^6 \, \text{N/m}\) (Incorrect unit, should be \(\text{N/m}^2\))
• Option 3: \(0.59 \times 10^6 \, \text{N/m}\) (Incorrect unit)
• Option 4: \(0.89 \times 10^6 \, \text{N/m}^2\)

Our calculated value of \(0.89 \times 10^6 \, \text{N/m}^2\) perfectly matches Option 4.