Question

A rectangular wire loop with length a and width b lies in the xy-plane as shown. Within the loop, there is a time dependent magnetic field given by $B = c [( x \cos \omega t ) \hat{ i }+( y \sin \omega t ) \hat{ k }]$ Here, $C$ and $\omega$ are constants. The magnitude of emf induced in the loop as a function of time is

• A. $\left| \frac{ab^2 c}{2} \, \omega \, \cos \, \omega t \right|$
• B. $\left| ab^2 c \, \omega \, \cos \, \omega t \right|$
• C. $\left| \frac{ab^2 c}{2} \, \omega \, \sin \, \omega t \right|$
• D. None of the options

Answer 1

The Correct Option is A

Area vector of the given loop, \(A = ab \hat{ k }\) Time dependent magnetic field, \(B = c [( x \cos \omega t ) \hat{ i }+(y \sin \omega t ) \hat{ k }]\) \(\therefore\) Magnetic flux through the loop, \(\phi_{n}=A \cdot B\) \(=a b \hat{k}[c\{(x \cos \omega t) \hat{i}+(y \sin \omega t) \hat{k}\}]\) \(=abcy sin \omega t\)...(i) \(\because y \leq y \leq b\) \(\therefore y_{a v}=\frac{0+b}{2}=\frac{b}{2}\) Putting the value of \(y\) in E (i), we get \(\therefore \phi_{ n }= abc \cdot \frac{ b }{2} \cdot \sin \omega t =\frac{ ab ^{2}}{2} csin \omega t\) \(\therefore\) Induced emf in the loop, \(\varepsilon=\frac{ d }{ dt } \phi_{ n }=\frac{ d }{ dt } \cdot \frac{ ab ^{2} c }{2} \sin \omega t\) Hence, \(|\varepsilon|=\left|\frac{ ab ^{2} c }{2} \omega \cos \omega t \right|\)

The magnetic impact on moving electric charges, magnetic materials, and electric currents is represented by the magnetic field, sometimes known as a vector field. Electrically charged particles in motion are compelled by magnetic fields to follow circular or helical paths, and they also encounter forces that are perpendicular to both their own velocities and the magnetic field. The region surrounding a magnet where magnetism is sensed can be described as a magnetic field.

The letters B or H can be used to represent the magnetic field's symbol. Mathematically, it is represented by quantities called vectors, which have both direction and magnitude. Magnetic flux density (also known as magnetic induction) and magnetic field strength (also known as magnetic field intensity), which are each represented by the vectors B and H, respectively, aid in the representation of the magnetic field.

The basic unit of a magnetic field is (Newton's Second)/Coulomb and the unit itself is called a tesla. It is well known that magnetic field lines don't cross one another. Magnetic lines really form complete loops that start at the north pole and conclude at the south pole. The strength of the field is often shown by the density of the field lines.