Answer:
[tex]a. \phi _B=3.0528\times10^-^3T\ m^2\\ \\b. \phi_B=1.83299\times10^-^3T\ m^2\\\\c.\phi_B=0[/tex]
Explanation:
#Consider a circular area of radius [tex]R=2.98cm[/tex] in the xy-plane at z=0. This means all the are vector points toward the +ve z-axis.
a. first, find the magnetic flux if the magnetic field has a magnitude of [tex]B=0.23T[/tex] and points toward the +ve z-axis. The angle between the magnetic field and the area is [tex]\theta=0[/tex]. Hence the magnetic flux:-
[tex]\phi _B=\int {\bar B} . d\bar A \\=\int BdAcos(\theta)=BAcos(0)=BA\\\\=\pi R^2B=\pi(6.50\times10^-^3m)^2(0.230T)\\=3.0528\times10^-^3 T\ m^2[/tex]
Hence flux magnitude in [tex]+z[/tex] direction is [tex]3.0528\times10^-^3T \ m^2[/tex]
b. We now find the magnetic flux when the field has a magnitude of B=0.230T and points at an angle of [tex]\theta=53.1\textdegree[/tex] from the [tex]+z[/tex] direction.
Magnetic flux is calculated as:
[tex]\phi _B=\int\bar B \bar dA\\=\int BdAcos (\theta)=BAcos(0)=BA\\=\pi R^2B=\pi(6.50\times 10^-^2m)^2(0.230T)\\=1.83299\times 10^-^3 T \ m^2[/tex]
Hence the flux at an angle of [tex]53.1\textdegree[/tex] is [tex]1.83299\times 10^-^3T \ m^2[/tex]
c. We now need to find the magnetic flux if the field has a magnitue of B=0.230T and points in the direction of +y-direction. As with the previous parts, the magnetic flux will be calculated as:
[tex]\phi_B= \int\bar B \times d\bar A\\=\int BdAcos(\theta)\\=BAcos(90\textdegree)\\=0[/tex]
Hence the magnetic flux in the +y-direction is zero.
(a) The magnetic flux in the positive z direction is 3.059 x 10⁻³ Wb.
(b) The magnetic flux at 53.1 degrees in positive z direction is 1.837 x 10⁻³ Wb.
(c) The magnetic flux in the positive y direction is 0 Wb.
The magnetic flux through the circle is calculated as follows;
Ф = BAcosθ
where;
A = πr²
A = π x (0.065)²
A = 0.0133 m³
Ф = 0.23 x 0.0133 x cos(0)
Ф = 3.059 x 10⁻³ Wb
Ф = 0.23 x 0.0133 x cos(53.1)
Ф = 1.837 x 10⁻³ Wb
Ф = 0.23 x 0.0133 x cos(90)
Ф = 0 Wb
Learn more about magnetic flux here: https://brainly.com/question/10736183
