Question

A flat, square surface with side length 3.40 cm is in the xy-plane at \(z =\) 0. Calculate the magnitude of the flux through this surface produced by a magnetic field \(\overrightarrow{B} =\) (0.200 T)\(\hat{\imath}\) + (0.300 T)\(\hat{\jmath}\) - (0.500 T)\(\hat{k}\).

Short answer

The magnitude of the magnetic flux is \(0.578 \times 10^{-3}\) Weber.

Step-by-step solution

Understand the Problem

We need to calculate the magnetic flux through a square surface with side 3.40 cm. The magnetic field vector is given as \(\overrightarrow{B} = (0.200) \hat{\imath} + (0.300) \hat{\jmath} - (0.500) \hat{k}\), and the square lies in the xy-plane, making its normal vector \(\hat{k}\).

Calculate the Area of the Square

The side length of the square is 3.40 cm. Converting it to meters gives us \(3.40 \times 10^{-2}\) m. The area \(A\) of the square is given by:\[A = (3.40 \times 10^{-2} \text{ m})^2 = 1.156 \times 10^{-3} \text{ m}^2\]

Determine the Relevant Component of Magnetic Field

Since the surface is on the xy-plane, the normal vector is \(\hat{k}\). Only the component of \(\overrightarrow{B}\) along \(\hat{k}\) contributes to the flux. Thus, the relevant component is: \[-0.500 \text{ T} \hat{k}\]

Calculate the Magnetic Flux

Magnetic flux \(\Phi_B\) through the surface is given by the dot product of the magnetic field and the area vector \(\overrightarrow{A}\), which points in the \(\hat{k}\) direction: \[\Phi_B = \overrightarrow{B} \cdot \overrightarrow{A} = (0.200 \hat{\imath} + 0.300 \hat{\jmath} - 0.500 \hat{k}) \cdot (1.156 \times 10^{-3} \hat{k}) = -0.500 \times 1.156 \times 10^{-3} \]\[\Phi_B = -0.578 \times 10^{-3} \text{ Wb}\]

Magnitude of the Flux

The magnitude of the flux is the absolute value of the calculated flux, which is:\[|\Phi_B| = 0.578 \times 10^{-3} \text{ Wb}\]

Key concepts

Understanding Magnetic Fields

Magnetic fields are invisible forces that exert influence on moving electric charges, magnetic material, and currents. These fields are represented by vector quantities, meaning they have both magnitude and direction.
In the exercise, the magnetic field \(\overrightarrow{B}\) is given in terms of its components along the x, y, and z axes: \(\overrightarrow{B} = (0.200) \hat{\imath} + (0.300) \hat{\jmath} - (0.500) \hat{k}\). This means:

• 0.200 Tesla in the x-direction (\(\hat{\imath}\)),
• 0.300 Tesla in the y-direction (\(\hat{\jmath}\)), and
• -0.500 Tesla in the z-direction (\(\hat{k}\)).

The negative sign indicates the direction of the field component is opposite to the positive axis.
Understanding the vector form of a magnetic field is essential for solving problems that involve magnetic flux, as it determines which component influences the flux through a given surface.

Flux Calculation Basics

Magnetic flux indicates how much of the magnetic field passes through a given surface. The key to calculating magnetic flux is using the dot product of the magnetic field vector \(\overrightarrow{B}\) and the area vector \(\overrightarrow{A}\).
The equation for magnetic flux \(\Phi_B\) is: \[\Phi_B = \overrightarrow{B} \cdot \overrightarrow{A}\] Where:

• \(\overrightarrow{B}\) is the magnetic field vector,
• \(\overrightarrow{A}\) is the area vector which is perpendicular to the surface,
• The \(\cdot\) represents the dot product, which multiples magnitudes of the vectors and the cosine of the angle between them.

For a surface oriented in the xy-plane, only the z-component of \(\overrightarrow{B}\) (i.e., along \(\hat{k}\)) would contribute to the flux since the area vector points in that direction.
By aligning the surface area vector correctly with the magnetic field vector, the flux is simplified to: \[\Phi_B = B_z A\] where \(B_z\) is the z-component of the magnetic field.

Role of Surface Area in Flux

The surface area plays a crucial role in determining how much of the magnetic field penetrates it. For a flat surface, such as a square, the area vector points perpendicular to the plane of the square.
In the given exercise, the square lies in the xy-plane, making the normal vector point in the z-direction. This means it only intercepts the magnetic field component aligned with this normal vector.
The area \(A\) of the square is calculated using the formula: \[A = \text{side length}^2\] Since the side of the square is given in centimeters, converting it to meters is necessary to keep it consistent with other units.

• First convert side length: \(3.40 \times 10^{-2} \text{ m}\)
• Calculate area: \[A = (3.40 \times 10^{-2} \text{ m})^2 = 1.156 \times 10^{-3} \text{ m}^2\]

This calculated area is then used to find the magnetic flux through the surface. The correct measurement of surface area is essential for obtaining accurate flux values.

Applying Vector Analysis in Flux Problems

Vector analysis simplifies understanding physical phenomena involving directions and magnitudes, like magnetic fields and flux. A fundamental aspect of vector analysis in flux calculations is understanding dot products.
The dot product \(\overrightarrow{B} \cdot \overrightarrow{A}\) simplifies to: \[B_x A_x + B_y A_y + B_z A_z\] Each vector component is multiplied and summed to give a scalar quantity representing the flux through the surface.

• Since \(\overrightarrow{A}\) is perpendicular to the plane, its only non-zero component is along \(\hat{k}\),
• Thus, the flux is influenced only by the \(\hat{k}\) component of \(\overrightarrow{B}\), reducing the equation to: \[\Phi_B = B_z A_z\]

Understanding the role of each vector component and how the dot product helps align the vectors correctly is crucial. It enables us to focus on the relevant components affecting the magnetic flux through a surface.
Efficiently using vector analysis can make solving complex physics problems straightforward and logical.