Question

I

I The metal equilateral triangle in FIGURE EX$30.11,20\mathrm{cm}$on each side, is halfway into a$0.10\mathrm{T}$magnetic field.

a. What is the magnetic flux through the triangle?

b. If the magnetic field strength decreases, what is the direction of the induced current in the triangle?

Short answer

(a) The magnetic flux is$\text{(a)}{\mathrm{\Phi }}_m=8.6\times {10}^{-4}\mathrm{Wb}\text{.}$

(b) The induced current flows clockwise.

Step-by-step solution

Step1:Definition of magnetic field

There is a magnetic field. Magnetic fields are vector fields that describe the magnetic influence of currents and magnetised materials. Magnetic fields are frequently seen in everyday life in the form of permanent magnets, which attract or repel other magnets by pulling on magnetic materials (such as iron).

step2:Find height(part a)

(a) Magnetic flux is the amount of magnetic field that flows through a loop in area A. Is when magnet is parallel to the normal of a plane, the magnetic flux is given

${\mathrm{\Phi }}_m=BA$

The triangle's area is equal to half of its base multiplied by its height. The height is derived from

$h=\left(20\mathrm{cm}\right)\sin 60=17.3\mathrm{cm}$

So, we get the area of the triangle by

$A=\frac{1}{2}bh=\frac{1}{2}\left(0.20\mathrm{m}\right)\left(17.3\times {10}^{-2}\mathrm{m}\right)=1.73\times {10}^{-2}{\mathrm{m}}^2$

step3:Find area of the flux

Because the magnetic field flows through half of the total loop, the flux area is

$A_{\mathrm{\Phi }}=\frac{1}{2}\left(1.73\times {10}^{-2}{\mathrm{m}}^2\right)=8.65\times {10}^{-3}{\mathrm{m}}^2$

Now, we plug the values forlocalid="1648879727201" $A_{\Phi },\theta =0$and B into equation (2) to get ${\Phi }_m$

${\mathrm{\Phi }}_m=B{A}_{\mathrm{\Phi }}\cos \theta =\left(0.10\mathrm{T}\right)\left(8.65\times {10}^{-3}{\mathrm{m}}^2\right)\cos 0^{\circ }$

Find direction(part b)

(b) According to Lenz's law, the direction of the induced emf drives current around a wire loop in the opposite direction of the change in magnetic flux that causes the emf. When the applied magnetic field's strength is reduced, the induced magnetic field points in the same direction as the applied magnetic field B. As a result, the induced magnetic field is pointing into the page.

To determine the direction of the induced current, apply the right-hand rule. Your four wrapped fingers represent the direction of the induced magnetic field (into the page), and your thumb represents the direction of the current.

$\text{(a)}{\mathrm{\Phi }}_m=8.6\times {10}^{-4}\mathrm{Wb}\text{.}$

(b) The induced current flows clockwise.