A metallic wire bend in the form a semi-circle of radius 0.1 m is moved in direction parallel to its plane, But perpendicular to a magnetic field b=20mT with a velocity of 10m/s. What is the induced emf in the wire?
Im stuck at finding $\frac {dA}{dt}$ of the semi-circle.It should be 2$\pi$rv but the books says its 2rv.
Help appreciated.
The electric current flows through a loop (drawn in red colour), which is the border of the area $A$.
The circuit is made up of an upper stationary wire, a lower stationary wire (both drawn in black) and the moving semic-circle shaped wire (the red border of the semicircle):
$A$ is the sum of the area of a growing rectangle (the shaded area in the image above) and the constant semi-circle area (not shaded).
We can write $A$ as $$ A = v t \cdot 2 r + \pi r $$
An animated version can be found here.
Then the derivative of $A$ regarding time $t$ is $$ \dot{A} = 2rv $$ Update: I assume a setup like this (with a semi-circular bar, not a straight one):
(Source Wikipedia)
