A wire bent as a parabola `y=ax^(2)` is located in a uniformed magnetic field of induaction `B` , the vector `B` being perpendicular to the plane `x-y` . At moment `t=0` a connector starts sliding translationwise from the parabola apex with a constant acceleration `omega` . Find the emf of electromagnetic induction in the loop thus formed as a function of `y`

Answer

To solve the problem step by step, let's break down the process of finding the induced electromotive force (emf) in the given scenario.

### Step 1: Understand the Setup
We have a wire bent in the shape of a parabola described by the equation \( y = ax^2 \). The wire is placed in a uniform magnetic field \( \mathbf{B} \), which is perpendicular to the x-y plane. A connector starts sliding from the apex of the parabola with a constant acceleration \( \omega \).

### Step 2: Determine the Position of the Connector
At time \( t \), the distance \( y \) covered by the connector can be expressed using the equation of motion:
\[ y = \frac{1}{2} \omega t^2 \]
From this, we can solve for \( t \):
\[ t = \sqrt{\frac{2y}{\omega}} \]

### Step 3: Calculate the Velocity of the Connector
The velocity \( v \) of the connector at time \( t \) can be calculated using:
\[ v = \omega t \]
Substituting the expression for \( t \):
\[ v = \omega \sqrt{\frac{2y}{\omega}} = \sqrt{2\omega y} \]

### Step 4: Find the Length of the Wire in Motion
The effective length \( L \) of the wire that is moving is determined by the parabola's equation. From \( y = ax^2 \), we can express \( x \) in terms of \( y \):
\[ x = \sqrt{\frac{y}{a}} \]
Thus, the effective length of the connector is:
\[ L = 2x = 2\sqrt{\frac{y}{a}} \]

### Step 5: Apply Faraday's Law of Electromagnetic Induction
The induced emf \( E \) can be calculated using the formula:
\[ E = vBL \]
Substituting the expressions for \( v \) and \( L \):
\[ E = \left(\sqrt{2\omega y}\right) B \left(2\sqrt{\frac{y}{a}}\right) \]
This simplifies to:
\[ E = 2B \sqrt{2\omega} \sqrt{y} \sqrt{\frac{y}{a}} = \frac{2B}{\sqrt{a}} \sqrt{2\omega} y \]

### Step 6: Final Expression for Induced EMF
Combining the constants, we can express the induced emf as:
\[ E = \frac{2B}{\sqrt{a}} \sqrt{2\omega} y \]
This is the required expression for the emf of electromagnetic induction in the loop as a function of \( y \).