Answer:
see explanation
Explanation:
Magnetic flux is defined by [tex]\Phi = \int \vec{B} \cdot d\vec{A}[/tex]
we have a disk with cross-section area [tex]A = \pi r^2[/tex], where [tex]r[/tex] is the radius of the disk ( r = 0.01 [m] ).
a) disk is perpendicular to the magnetic field
We assume the magnetic field is coming from the bottom, therefore [tex]\vec{B}[/tex] and [tex]d\vec{A}[/tex] are parallel and the dot product is maximum because the angle between the vectors is 0.
the magnetic flux takes the following form:
[tex]\Phi = \int \vec{B} \cdot d\vec{A}=\int B*dA*cos(0)=\int B*dA[/tex]
now the magnitude of B is constant, we have:
[tex]B\int dA[/tex]
remember what [tex]A[/tex] is? then we just derivate it with respect to the radius and we get
[tex]dA = \frac{d }{dr}(\pi r^2)=2\pi rdr[/tex]
the flux now is [tex]\Phi=\B\int dA=B \int 2\pi rdr=2\pi B\int rdr= 2\pi B(\frac{r^2}{2})=\pi Br^2[/tex]
we just demonstrated that the flux of a magnetic field whose direction and magnitude are constant is equal to B times the area A of the surface the magnetic field is passing through:
now we replace values
[tex]\Phi = 0.01 [T] * \pi *(0.01)^2=\pi *10^{-6}[/tex]
b) Now if the surface is oriented at 45° we go back a few steps and we just have a small difference this time:
[tex]\Phi = \int \vec{B} \cdot d\vec{A}=\int B*dA*cos(45)=\pi*B*r^2*cos(45)[/tex]
therefore
[tex]\Phi = 0.01 [T] * \pi *(0.01)^2*cos(45)=\pi *10^{-6}*\frac{1}{\sqrt{2}}[/tex]
the magnetic flux has decreased because of the incidence angle
c) if the surface vector is perpendicular to the magnetic field then the expression takes the following form:
[tex]\Phi = \int \vec{B} \cdot d\vec{A}=\int B*dA*cos(90)=\pi*B*r^2*0 = 0[/tex]
there is no magnetic flux because the thin disk is perpendicular to the magnetic field
