Answer:
The magnetic induction of the magnetic field is 0.0005293 mT
Explanation:
Data given
I = 7 A = the total current in the wire
r = 23 cm = the radius of the wire = 0.23 meter
r' = 2cm = the measurement point, which should be inside the wire = 0.02 meter
Let's consider the current density is constant in the wire, ⇒ the current enclosed is a function of the enclosed area
I(enclosed) = Jπ
r
²
we can consider the current density as the total current over the whole area:
I(enclosed) = I / (πr
²) * πr'
²
I(enclosed) = (I* r'²)/ (r
²)
with I = total current in the wire = 7A
With r = the radius of wire = 0.23 meter
with r' = the distance of point from the center of wire 0.02 meter
We plug this into ampere's law:
∮
*B
*dl =μ
0 * (I* r'²)/ (r
²)
with B = Magnetic flux density (in Tesla) or magnetic induction
with dl = an infinitesimal element (a differential) of the curve C
with µ0 = the magnectic constant = 4π*10^−7 H/m
We can simplify this, by using an Amperian loop can write this as:
B
*(
2
π
r') = μ
0 * (I* r'²)/ (r
²)
Because the circumference of a circle is 2
π
r
, when we integrate over length at a distance r
′ from the center of wire whose crossection is a circle we get 2
π
r
′
When we isolate B, we get:
B = µo *(Ir'/2
π
r
²)
B = 4π*10^−7 * ((7*0.02)/2*π*0.23²)
B =5.293 *10 ^-7 T = 0.0005293 mT
The magnetic induction of the magnetic field is 0.0005293 mT
