Problem 13 - University Physics with Modern Physics

Question

An open plastic soda bottle with an opening diameter of 2.5 \(\mathrm{cm}\) is placed on a table. A uniform \(1.75-\mathrm{T}\) magnetic field directed upward and oriented \(25^{\circ}\) from vertical encompasses the bottle. What is the total magnetic flux through the plastic of the soda bottle?

Step-by-Step Solution

Answer

The magnetic flux through the soda bottle is approximately \( 7.78 \times 10^{-4} \text{ Wb} \).

Step 1: Understand the Problem

We need to find the magnetic flux through the surface of the open plastic soda bottle. The magnetic flux is calculated as the product of the magnetic field, the area it penetrates, and the cosine of the angle between the field and the normal to the surface.

Step 2: Calculate the Area of the Opening

The opening of the bottle is a circle with a diameter of 2.5 cm. First, we convert this diameter into meters: \[ d = 2.5 \text{ cm} = 0.025 \text{ m} \] The radius \( r \) is half of the diameter:\[ r = \frac{d}{2} = 0.0125 \text{ m} \]The area \( A \) of the circular opening is given by:\[ A = \pi r^2 = \pi (0.0125)^2 \approx 4.91 \times 10^{-4} \text{ m}^2 \]

Step 3: Calculate the Magnetic Flux

We use the magnetic flux formula, which is:\[ \Phi = B \cdot A \cdot \cos \theta \]Where:- \( B = 1.75 \text{ T} \) is the magnetic field strength.- \( A = 4.91 \times 10^{-4} \text{ m}^2 \) is the area of the opening calculated earlier.- \( \theta = 25^{\circ} \) is the angle from vertical. This means the angle between the magnetic field and the normal to the surface (which is vertical) is \( \theta \).Substituting the values:\[ \Phi = 1.75 \cdot 4.91 \times 10^{-4} \cdot \cos(25^{\circ}) \]Calculate \( \cos(25^{\circ}) \) using a calculator:\[ \cos(25^{\circ}) \approx 0.9063 \]Thus, the magnetic flux is:\[ \Phi \approx 1.75 \cdot 4.91 \times 10^{-4} \cdot 0.9063 \approx 7.78 \times 10^{-4} \text{ Wb} \]

Step 4: Conclusion

The total magnetic flux through the plastic of the soda bottle is approximately \( 7.78 \times 10^{-4} \text{ Wb} \).

Key Concepts

Magnetic Field

A magnetic field is a region around a magnetic material or a moving electric charge within which the force of magnetism acts. Magnetic fields are vector fields, usually denoted by the symbol \( B \), and are measured in teslas (T) or gauss in the CGS system. In this exercise, the magnetic field has a strength of \( 1.75 \) T, pointing upwards but slightly tilted. Understanding the direction and strength of a magnetic field is crucial when evaluating its interaction with surfaces, such as determining magnetic flux through a bottle's opening.

Area Calculation

Area calculation is an important step when determining magnetic flux through a surface. It is the surface area that the magnetic field lines penetrate. In this problem, the soda bottle’s opening is a circle. To calculate its area, you first need the diameter, which is given as \( 2.5 \) cm. Convert it into meters for consistency with SI units: \( d = 0.025 \) m.

• Radius \( r \) is \( \frac{d}{2} \).
• The area \( A \) of a circle is given by \( \pi r^2 \).

After doing the math, you find \( A = 4.91 \times 10^{-4} \) m\(^2\). This value is key in calculating the magnetic flux.

Cosine Function

The cosine function plays a significant role in calculating magnetic flux because it accounts for the angle between the magnetic field and the normal (perpendicular) to the surface. This exercise uses \( \cos(25^{\circ}) \), where \( 25^{\circ} \) is the given angle of inclination.

• The cosine function ranges from -1 to 1.
• It modifies the magnetic field's effect based on angle, reducing it if not perfectly aligned.

For \( 25^{\circ} \), \( \cos(25^{\circ}) \approx 0.9063 \), which is used in the formula:
\( \Phi = B \cdot A \cdot \cos \theta \). This function determines the proportion of the magnetic field that directly penetrates the surface.

Angle of Inclination

The angle of inclination defines the tilt between the magnetic field and the normal (perpendicular) to the surface. Here, the field is inclined at \( 25^{\circ} \) from vertical. This concept is crucial because the magnetic flux depends not just on the field strength and area, but also the field's orientation relative to the surface.

• An angle of \( 0^{\circ} \) would mean full magnetic flux with \( \cos(0) = 1 \).
• As the angle increases, \( \cos \theta \) decreases, reducing effective field penetration.

Understanding this angle helps predict how much of the magnetic field contributes to the actual flux through the surface.