Problem 11 - University Physics with Modern 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 \(B=0.230 \mathrm{T}\) (a) in the \(+z\) -direction; \((b)\) at an angle of \(53.1^{\circ}\) from the \(+z\) -direction; \((\mathrm{c})\) in the \(+y\) -direction?

Step-by-Step Solution

Answer

(a) 0.00306 T⋅m²; (b) 0.00184 T⋅m²; (c) 0 T⋅m².

Step 1: Understand Magnetic Flux

Magnetic flux (\(\Phi\) ) through a surface is given by the formula \(\Phi = B \cdot A \cdot \cos(\theta)\), where \(B\) is the magnetic field, \(A\) is the area, and \(\theta\) is the angle between the magnetic field and the normal (perpendicular) to the surface.

Step 2: Calculate the Area of the Circle

Given the radius \(r = 6.50\, \mathrm{cm}\), convert it to meters: \(r = 0.0650\, \mathrm{m}\). The area \(A\) of a circle is \(A = \pi r^2\). Substituting the radius, we get \(A = \pi \, (0.0650)^2 = 0.0133 \, \mathrm{m^2}\).

Step 3: Calculate Magnetic Flux for Case (a)

For case (a), the magnetic field is perpendicular to the plane of the circle (\(\theta = 0^{\circ}\)). Thus, \(\cos(0^{\circ}) = 1\). Substituting the values into the flux formula \(\Phi = 0.230 \times 0.0133 \times 1 = 0.00306 \, \mathrm{T\cdot m^2}\).

Step 4: Calculate Magnetic Flux for Case (b)

For case (b), the angle \(\theta = 53.1^{\circ}\). Calculate \(\cos(53.1^{\circ})\) which is approximately \(0.6018\). Substituting into the formula \(\Phi = 0.230 \times 0.0133 \times 0.6018 = 0.00184 \, \mathrm{T\cdot m^2}\).

Step 5: Calculate Magnetic Flux for Case (c)

For case (c), the magnetic field is parallel to the plane of the circle (\(\theta = 90^{\circ}\)). Thus, \(\cos(90^{\circ}) = 0\). Therefore, the magnetic flux is \(\Phi = 0.230 \times 0.0133 \times 0 = 0 \, \mathrm{T\cdot m^2}\).

Key Concepts

Magnetic Field

The magnetic field, often denoted by \( B \), is a vector field surrounding magnets and electric currents, and it exerts magnetic force on moving charges. The strength of a magnetic field is measured in teslas (\( \text{T} \)). It can have different orientations, and its relationship with a surface or object is crucial in determining magnetic effects.

• If the magnetic field is perpendicular to a plane, it maximizes the interaction causing the highest possible magnetic flux for a given magnitude of \( B \).
• If the magnetic field is parallel to the plane, the effect is minimized, as seen in cases like a magnetic field in the \( +y \)-direction with a surface lying on the \( xy \)-plane.
• Any intermediate angle between the field and the surface needs to be considered through its cosine function value, as used in the calculation of magnetic flux through non-perpendicular angles.

Understanding the orientation and interactions of magnetic fields helps in various applications from electrical engineering to geophysics.

Area of a Circle

The area of a circle is a fundamental concept in geometry, critical to calculating quantities like magnetic flux through circular surfaces. It is calculated using the formula: \[ A = \pi r^2 \] where \( r \) is the radius of the circle. This formula implies that the area grows with the square of the radius.

• For a radius of 6.50 cm, converting to meters is necessary for consistency with standard units in physics, giving us \( r = 0.0650 \ \text{m} \).
• Plugging this into the formula yields an area of approximately \( A = 0.0133 \ \mathrm{m^2} \).

When calculating the magnetic flux, knowing the accurate area is critical, since any error can lead to misinterpretation of magnetic interactions.

Angle in Physics

In physics, angles play a crucial role in understanding how forces and fields interact with surfaces. The angle \( \theta \) is particularly important in calculations of magnetic flux, where it defines the orientation between a magnetic field and the normal to a surface.

• When the angle is \( 0^{\circ} \), the magnetic field is perpendicular to the plane, leading to maximum flux through the surface since \( \cos(0^{\circ}) = 1 \).
• At an angle of \( 90^{\circ} \), the field is parallel to the surface, and the magnetic flux equals zero because \( \cos(90^{\circ}) = 0 \).
• Intermediate angles require the cosine of the angle, such as \( \cos(53.1^{\circ}) \approx 0.6018 \) in our example, to calculate the effective contribution of the magnetic field to the flux.

The cosine function captures how much of the field is "effective" in generating flux over the area of interest. Thus, angle measurement and calculations ensure that magnetic interaction assessments are precise.