Problem 10 - University Physics with Modern Physics

Question

The magnetic flux through one face of a cube is \(+0.120 \mathrm{Wb} .\) (a) What must the total magnetic flux through the other five faces of the cube be? (b) Why didn't you need to know the dimensions of the cube in order to answer part (a) 2 (c) Suppose the magnetic flux is due to a permanent magnet like that shown in Fig. \(27.11 .\) In a sketch, show where the cube in part (a) might be located relative to the magnet.

Step-by-Step Solution

Answer

(a) -0.120 Wb; (b) Gauss's Law; (c) Near the north pole of the magnet.

Step 1: Understanding Magnetic Flux and Gauss's Law

In a closed surface, the net magnetic flux is always zero. This is Gauss's Law for magnetism, which states that the total magnetic flux through any closed surface is zero. This implies that the magnetic field lines entering a closed surface must equal the lines exiting it.

Step 2: Applying Gauss's Law to the Cube

For the cube, the net magnetic flux through all six faces must add up to zero. Given one face has a flux of \(+0.120\, \mathrm{Wb}\), the sum of the flux through the other five faces must be \(-0.120\, \mathrm{Wb}\) to ensure the total flux is zero.

Step 3: Non-reliance on Cube Dimensions

Gauss's Law for magnetism doesn't depend on the size or shape of the object. It applies equally to any closed surface regardless of its dimensions. Thus, the dimensions of the cube are irrelevant for calculating total magnetic flux.

Step 4: Placing the Cube Relative to the Magnet

To sketch the cube's placement, consider that magnetic field lines exit from the north pole and enter the south pole of a magnet. Position the cube such that its face with positive flux aligns with exiting field lines. The sketch would show the cube near the north pole of the magnet, aligning one face with the exiting lines.

Key Concepts

Magnetic Flux

Magnetic flux is a measure of the quantity of magnetism, taking into account the strength and extent of a magnetic field. It is denoted in Weber (Wb).
Magnetic flux through a surface is calculated by multiplying the magnetic field strength (\( B \) ) by the area (\( A \) ) the field penetrates, and the angle (\( \theta \) ) between the field and the normal (perpendicular) to the surface: \(\Phi = B \times A \times \cos(\theta)\).
This concept is important because a change in magnetic flux can induce an electromotive force (emf) in nearby conductors, which is the principle behind many electrical generators and transformers. In the exercise, the cube has six faces, and the magnetic flux through one of these faces is given as \( +0.120 \text{ Wb} \). The task is to calculate the total magnetic flux through the other five faces.

Closed Surface

A closed surface is a surface that completely encloses a space, much like a sphere or a cube. In the context of Gauss's Law for magnetism, it is crucial because it clarifies how magnetic fields behave.
According to Gauss's Law for Magnetism, the net magnetic flux through a closed surface is always zero. This means that the total magnetic flux entering the surface equals the flux exiting the surface.
This concept is helpful to answer question (a) in the exercise, where, given a flux through one face of the cube, the flux through the remaining five faces must cumulatively negate it to satisfy Gauss's Law, resulting in a net flux of zero through the cube.

Cube Placement

The placement of the cube in magnetic fields is influenced by how magnetic flux enters and exits the magnet. When dealing with a permanent magnet, field lines emerge from the north pole and loop around to enter the south pole. The exercise hints at placing the cube such that one face receiving \( +0.120 ext{ Wb} \) from the magnetic field might be aligned with these field lines exiting from the north pole.
By understanding the distribution of magnetic field lines around a magnet, we can position the cube to appropriately line up one face for optimal observation of magnetic flux effects. This mental model assists in visualizing the roles of different cube faces and how magnetic fields interact with them.