Problem 1 - College Physics

Question

A circular area with a radius of 6.50 \(\mathrm{cm}\) lies in the \(x\) -y plane. What is the magnitude of the magnetic flux through this circle due to a uniform magnetic field \(B=0.230 \mathrm{T}\) that points (a) in the \(+z\) direction? (b) at an angle of \(53.1^{\circ}\) from the \(+z\) direction? (c) in the \(+y\) direction?

Step-by-Step Solution

Answer

(a) 0.00306 Wb, (b) 0.00184 Wb, (c) 0 Wb.

Step 1: Calculate Area of the Circle

First, calculate the area of the circular area. The formula for the area of a circle is \(A = \pi r^2\), where \(r\) is the radius of the circle. Here, \(r = 6.50 \, \mathrm{cm} = 0.065 \, \mathrm{m}\). Substituting, we have:\[A = \pi (0.065)^2 \approx 0.0133 \, \mathrm{m^2}\]

Step 2: Calculate Magnetic Flux for (a) - Field in +z Direction

The magnetic flux \(\Phi\) for a magnetic field \(B\) perpendicular to the surface (along the normal) is given by \(\Phi = B \cdot A\). Since the magnetic field is in the \(+z\) direction and the circle is in the \(x-y\) plane, the angle \(\theta\) between them is \(0^\circ\). Thus, \[ \Phi = B \cdot A \cdot \cos(0^\circ) = 0.230 \, \mathrm{T} \times 0.0133 \, \mathrm{m^2} \approx 0.00306 \, \mathrm{Wb}\]

Step 3: Calculate Magnetic Flux for (b) - Field at 53.1° from +z Direction

In this case, the angle \(\theta\) between the magnetic field and the normal to the circle is \(53.1^\circ\). Thus, the magnetic flux is calculated as:\[ \Phi = B \cdot A \cdot \cos(53.1^\circ) = 0.230 \, \mathrm{T} \times 0.0133 \, \mathrm{m^2} \times \cos(53.1^\circ) \approx 0.00184 \, \mathrm{Wb}\]

Step 4: Calculate Magnetic Flux for (c) - Field in +y Direction

When the magnetic field is in the \(+y\) direction, it is perpendicular to the normal of the circular area, as the normal is in the \(+z\) direction. Therefore, the angle \(\theta = 90^\circ\). The magnetic flux is:\[ \Phi = B \cdot A \cdot \cos(90^\circ) = 0.230 \, \mathrm{T} \times 0.0133 \, \mathrm{m^2} \times 0 = 0 \, \mathrm{Wb} \]

Key Concepts

Magnetic Field

The concept of the magnetic field is central in understanding how forces affect charged particles in motion. A magnetic field represents a region where a magnetic force can be detected. It is often generated by moving electric charges or magnetic dipoles and can be uniform or vary in strength and direction.
Magnetic field lines provide a way to visualize a magnetic field. They start from the north pole and end at the south pole of a magnet. The direction of these lines indicates the direction of the magnetic force a north pole of a magnet would experience. The density of these lines reflects the strength of the magnetic field: the closer the lines, the stronger the field.

• The magnetic field can be measured in units called Tesla (T).
• A uniform magnetic field means the magnetic lines are parallel and equally spaced.

In the context of calculating magnetic flux through a circle, the field could have different orientations relative to the circle, affecting the amount of flux.

Circular Area

A circular area is simply the two-dimensional space enclosed by a circle. The size of this area plays an important role when calculating magnetic flux. The larger the area, the more flux it can capture, assuming other things like the magnetic field strength and orientation remain constant.
The formula for determining the area of a circle is given by:\[A = \pi r^2\]where \(r\) is the radius. If it's essential to work in consistent units, remember to convert all measurements to meters or the standard unit you're using.

• Area is measured in square meters \( \mathrm{m^2} \) in physics.
• A precise measurement is crucial for accurate calculations.

By knowing the circle's area, you can calculate how much of a magnetic field passes through it, when combined with the properties of the magnetic field and its angle of inclination.

Angle of Inclination

The angle of inclination, expressed as \( \theta \), is the angle between the magnetic field line and the normal (perpendicular) to the surface of the circular area. This angle is crucial because it affects the magnitude of the magnetic flux.
Magnetic flux \( \Phi \) through a surface is calculated by the formula:\[\Phi = B \cdot A \cdot \cos(\theta)\]where \(B\) is the magnetic field strength, \(A\) is the area, and \(\theta\) is this angle of inclination.

• When the field is perpendicular to the surface (\(\theta = 0^\circ \)), \(\cos(0^\circ) = 1\), indicating maximum flux.
• When parallel (\(\theta = 90^\circ\)), \(\cos(90^\circ) = 0\), meaning no flux passes through.

The inclination angle tells us how much of the magnetic field actually penetrates the surface, effectively tuning the amount of flux through geometric orientation.