Respuesta :
Answer:
[tex]\frac{di}{dt} = 7.31 \ A/s[/tex]
Explanation:
From the question we are told that
The number of turns is [tex]N = 450 \ turns[/tex]
The radius is [tex]r = 1.17 \ cm = 0.0117 \ m[/tex]
The position from the center consider is x = 3.45 cm = 0.0345 m
The induced emf is [tex]e = 8.20 *10^{-6} \ V/m[/tex]
Generally according to Gauss law
[tex]\int\limits { e } \, dl = \mu_o * N * \frac{di}{dt } * A[/tex]
=> [tex]e * 2\pi x = \mu_o * N * \frac{d i }{dt } * A[/tex]
Where A is the cross-sectional area of the solenoid which is mathematically represented as
[tex]A = \pi r ^2[/tex]
=> [tex]e * 2\pi x = \mu_o * N * \frac{d i }{dt } * \pi r^2[/tex]
=> [tex]\frac{di}{dt} = \frac{2e * x }{\mu_o * N * r^2}[/tex]ggl;
Here [tex]\mu_o[/tex] is the permeability of free space with value
[tex]\mu_o = 4\pi * 10^{-7} \ N/A^2[/tex]
=> [tex]\frac{di}{dt} = \frac{2 * 8.20*10^{-6} * 0.0345 }{ 4\pi * 10^{-7} * 450 * (0.0117)^2}[/tex]
=> [tex]\frac{di}{dt} = 7.31 \ A/s[/tex]
The value of di/dt from the given values of the solenoid electric field is;
di/dt = 7.415 A/s
We are given;
Number of turns; N = 450 per m
Radius; r = 1.17 cm = 0.0117 m
Electric Field; E = 8.2 × 10⁻⁶ V/m
Position of electric field; r' = 3.45 cm = 0.0345 m
According to Gauss's law of electric field;
∫| E*dl | = |-d∅/dt |
Now, ∅ = BA = μ₀niA
where;
n is number of turns
i is current
A is Area
μ₀ = 4π × 10⁻⁷ H/m
Thus;
E(2πr') = (d/dt)(μ₀niA) (negative sign is gone from the right hand side because we are dealing with magnitude)
Since we are looking for di/dt, then we have;
E(2πr') = (di/dt)(μ₀nA)
Making di/dt the subject of the formula gives;
di/dt = E(2πr')/(μ₀nA)
Plugging in the relevant values gives us;
di/dt = (8.2 × 10⁻⁶ × 2 × π × 0.0345)/(4π × 10⁻⁷ × 450 × π × 0.0117²)
di/dt = 7.415 A/s
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