Question

An electric field of 100 V/m is directed outward from the plane of a circular area with radius 4.0 cm. If the electric field increases at a rate of 10 V/ms, determine the magnitude of the magnetic field at a radial distance 10.0 cm away from the center of the circular area. 1.1 × 10-15 T 3.4 × 10-15 T 5.9 × 10-15 T 8.9 × 10-15 T 9.7 × 10-15 T

Answer 1

The magnitude of the magnetic field is
`8.9*10^(-19)\ T`

Step-by-step explanation:

Given that,

Electric field = 100 V/m

Radius = 4.0 cm

Electric field increase at a rate = 10 V/ms

Radial distance = 10.0 cm

We need to calculate the magnetic field

Using Gauss's law

`\oint{\vec{E}\cdot\vec{dA}}=\phi_(E)`

`(dE)/(dt)A=(d\phi_(E))/(dt)`

`(dE)/(dt)(\pi r^2)=(d\phi_(E))/(dt)`

We need to calculate the
`(d\phi)/(dt)`

`(d\phi)/(dt)=10*\pi*(4.0*10^(-2))^2`

`(d\phi)/(dt)=0.0503\ Nm^2/C.s`

According to Ampere Maxwell law

`\oint{\vec{B}\cdot \vec{ds}}=\mu_(0)(I+\epsilon_(0)(d\phi_(E))/(dt))`

`\oint{\vec{B}\cdot\vec{ds}}=\mu_(0)I+\mu_(0)\epsilon_(0)(d\phi_(E))/(dt))`

Electric field is zero inside the circle.

`\oint{\vec{B}\cdot \vec{ds}}=\mu_(0)\epsilon_(0)(d\phi_(E))/(dt))`

`B(2\pi*10.0*10^(-2))=4\pi*10^(-7)*8.85*10^(-12)*0.0503`

`B=(4\pi*10^(-7)*8.85*10^(-12)*0.0503)/(2\pi*10.0*10^(-2))`

`B=8.9*10^(-19)\ T`

Hence, The magnitude of the magnetic field is
`8.9*10^(-19)\ T`