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Calculate Magnetic Flux Density of Air-Core Solenoid with 20,000 A/m Field Strength
Physics
Grade 11 of Senior High School
Question Content
Determine the magnetic flux density B of an air core solenoid if its magnetic field strength H is 20,000 Amp-turns per metre. Given: $\\frac{\\vec{B}}{\\vec{H}} = \\mu_0 \\mu_r$, permeability of free space $\\mu_0 = 4\\pi×10^{-7}$ H/m
Correct Answer
0.0251 T
Detailed Solution Steps
1
Step 1: Identify the properties of an air-core solenoid. For air, the relative permeability $\\mu_r = 1$, since air is a non-magnetic material and does not amplify the magnetic field.
2
Step 2: Recall the formula relating magnetic flux density (B), magnetic field strength (H), permeability of free space ($\\mu_0$) and relative permeability ($\\mu_r$): $B = \\mu_0 \\mu_r H$.
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Step 3: Substitute the given values into the formula: $\\mu_0 = 4\\pi×10^{-7}$ H/m, $\\mu_r = 1$, $H = 20000$ A/m.
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Step 4: Calculate the value: $B = 4\\pi×10^{-7} × 1 × 20000 = 8\\pi×10^{-3} ≈ 0.0251$ T.
Knowledge Points Involved
1
Magnetic Flux Density (B) and Magnetic Field Strength (H) Relationship
The relationship is defined by $B = \\mu_0 \\mu_r H$, where $B$ is the magnetic flux density (measured in Tesla, T), $H$ is the magnetic field strength (measured in Amp-turns per metre, A/m), $\\mu_0$ is the permeability of free space, and $\\mu_r$ is the relative permeability of the material. This formula describes how a material modifies the magnetic field produced by a given current distribution.
2
Permeability of Free Space ($\\mu_0$)
$\\mu_0 = 4\\pi×10^{-7}$ H/m (Henries per metre) is a fundamental physical constant representing the magnetic permeability of a perfect vacuum. It quantifies how easily a magnetic field can pass through empty space.
3
Relative Permeability ($\\mu_r$)
Relative permeability is the ratio of the permeability of a material to the permeability of free space ($\\mu_r = \\frac{\\mu}{\\mu_0}$). For non-magnetic materials like air, $\\mu_r = 1$, meaning they do not enhance or reduce the magnetic field. For ferromagnetic materials, $\\mu_r$ can be much larger than 1, making them effective at concentrating magnetic fields.
4
Air-core Solenoid Magnetic Properties
An air-core solenoid uses air as the core material instead of a ferromagnetic core. Since air has a relative permeability of 1, its magnetic flux density is directly determined by the product of the permeability of free space and the magnetic field strength of the solenoid.
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