In a region at a distance r from z-axis, magnetic field `vec(B) = B_(0)rt hat(k)` is present where `B_(0)` is constant and t is time. Then the magnetic of induced electric field at a distance r from z-axis is given by

Options

A. `r/2 B_(0)`
B. `(r^(2))/2 B_(0)`
C. `(r^(2))/3 B_(0)`
D. none

Explanation

To solve the problem of finding the induced electric field at a distance \( r \) from the z-axis due to a time-varying magnetic field, we can follow these steps:

### Step 1: Understand the Given Magnetic Field
The magnetic field is given by:
\[ \vec{B} = B_0 R t \hat{k} \]
where \( B_0 \) is a constant, \( R \) is the radial distance from the z-axis, and \( t \) is time.

### Step 2: Calculate the Magnetic Flux
The magnetic flux \( \Phi \) through a circular area of radius \( r \) is given by:
\[ \Phi = \int \vec{B} \cdot d\vec{A} \]
where \( d\vec{A} \) is the differential area vector. For a circular area in the plane perpendicular to the z-axis, \( d\vec{A} = dA \hat{k} \) and \( dA = 2\pi r \, dr \).

Substituting \( \vec{B} \) and \( dA \):
\[ \Phi = \int B_0 R t \hat{k} \cdot (2\pi r \, dr \hat{k}) = \int B_0 R t (2\pi r) \, dr \]
This simplifies to:
\[ \Phi = B_0 R t \cdot 2\pi \int r \, dr \]

### Step 3: Integrate to Find the Flux
Now, we need to integrate:
\[ \int r \, dr = \frac{r^2}{2} \]
Thus, the flux becomes:
\[ \Phi = B_0 R t \cdot 2\pi \left(\frac{r^2}{2}\right) = \pi B_0 R t r^2 \]

### Step 4: Differentiate the Flux with Respect to Time
To find the induced electric field, we need to differentiate the flux with respect to time \( t \):
\[ \frac{d\Phi}{dt} = \pi B_0 R r^2 \frac{d}{dt}(t) = \pi B_0 R r^2 \]

### Step 5: Use Faraday's Law of Induction
According to Faraday's law, the induced electromotive force (emf) is given by:
\[ \mathcal{E} = -\frac{d\Phi}{dt} \]
Thus:
\[ \mathcal{E} = -\pi B_0 R r^2 \]

### Step 6: Relate emf to the Electric Field
The induced emf can also be expressed as:
\[ \mathcal{E} = \oint \vec{E} \cdot d\vec{L} \]
For a circular path of radius \( r \), this becomes:
\[ \mathcal{E} = E \cdot (2\pi r) \]

### Step 7: Set the Two Expressions for emf Equal
Equating the two expressions for emf:
\[ E \cdot (2\pi r) = -\pi B_0 R r^2 \]

### Step 8: Solve for the Electric Field \( E \)
Dividing both sides by \( 2\pi r \):
\[ E = -\frac{B_0 R r^2}{2r} = -\frac{B_0 R r}{2} \]

### Final Result
The induced electric field at a distance \( r \) from the z-axis is:
\[ E = \frac{B_0 R r}{2} \]