Q30P - Fundamentals Of Physics

Question

: Assume the average value of the vertical component of Earth’s magnetic field is $\mathbf{43}\mathbf{ }\mathit{\mu }\mathbf{T}$  (downward) for all of Arizona, which has an area of ${\mathbf{\text{2.95×10}}}^{\mathbf{5}}\mathbf{ }{\mathbf{km}}^{\mathbf{2}}$. (a)What is the magnitude and (b) What is the direction (inward or outward) of the net magnetic flux through the rest of Earth’s surface (the entire surface excluding Arizona)?

Step-by-Step Solution

Answer

(a) The magnitude of net magnetic flux through the rest of the Earth’s surface is ${\varphi }_{earth}=1.3\times {10}^7 \text{Wb}$.

(b) Its direction is outward

Step 1: Given

The magnetic field of Arizona,  $B=43 \mu \text{T}$

Area of Arizona,  $A=2.95\times {10}^5 {\text{km}}^{\text{2}}$

Step 2: Determining the concept

Use the magnetic flux formula to find the net magnetic flux through the rest of the earth’s surface. The sign of the flux shows its direction. Magnetic flux is a measurement of the total magnetic field which passes through a given area. It is a useful tool for helping describe the effects of the magnetic force on something occupying a given area.

The formula is as follows:

 $\varphi =BA$

 

where $\varphi$ is the flux,  B is the magnetic field, and A  is the area. 

Step 3: (a) Determining the magnitude of net magnetic flux through the rest of the Earth’s surface

The magnitude of net magnetic flux through the rest of the Earth’s surface:

Take the negative sign for downward magnetic field and positive sign for the upward magnetic field.

The magnetic flux through Arizona is,

 ${\varphi }_B=-BA$

 ${\varphi }_B=\left(-43\times {10}^{-6} \mathrm{T}\right)\left(2.95\times {10}^5 {\mathrm{km}}^2\right){\left({10}^3 \frac{\mathrm{m}}{\mathrm{km}}\right)}^2$

 ${\varphi }_B=-1.3\times {10}^7 \text{Wb}$

The negative sign shows that this flux goes inward.

Since

  $\begin{array}{rcl}{\varphi }_{net}& =& 0\\ {\varphi }_{earth}+{\varphi }_{arizona}& =& 0\\ {\varphi }_{earth}& =& -\left({\varphi }_{arizona}\right)\\ {\varphi }_{earth}& =& -\left(-1.3\times {10}^7 \text{Wb}\right)\\ {\varphi }_{earth}& =& 1.3\times {10}^7 \text{Wb}\end{array}$

 This is the flux that goes through the rest of the earth’s surface.

Therefore, the magnitude of net magnetic flux through the rest of the Earth’s surface is $1.3\times {10}^7 \text{Wb}$.

Step 4: (b) Determining the direction (inward or outward) of the net magnetic flux through the rest of Earth’s surface (the entire surface excluding Arizona)

The positive sign of the earth’s flux shows that its direction is outward.

Hence, its direction is outward.