A uniform magnetic field B exists perpendicular to the plane of a circular region of radius R . The magnetic field is increasing at a constant rate d B / d t = α . What is the magnitude of the induced

Question

A uniform magnetic field B exists perpendicular to the plane of a circular region of radius R . The magnetic field is increasing at a constant rate d B / d t = α . What is the magnitude of the induced electric field at a distance r ( r > R ) from the center of the circular region?

TestHub · Accepted Answer

According to Faraday's law in integral form, $\oint E \cdot d l = - \frac{d Φ _{B}}{d t}$ .

For a circular path of radius $r > R$ , the induced electric field $E$ is tangential and uniform due to symmetry. So, $\oint E \cdot d l = E (2 π r)$ . The magnetic flux through the circular region of radius $R$ is $Φ_{B} = B (π R^{2})$ . So, $\frac{d Φ _{B}}{d t} = (\frac{d B}{d t}) (π R^{2}) = α π R^{2}$ . Therefore, $E (2 π r) = α π R^{2}$ , which gives $E = \frac{α R ^{2}}{2 r}$ .

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