The magnetic flux linked with a closed coil is increased to a maximum value in $2 \mathrm{~s}$ and its…
The magnetic flux linked with a closed coil is increased to a maximum value in $2 \mathrm{~s}$ and its relation with time is $\phi=\mathrm{at}^2+\mathrm{bt}+\mathrm{c}$ then the relation between $\mathrm{a}, \mathrm{b}$ and $\mathrm{c}$ is
$a=-b$
$a=\frac{-b}{4}$
$a+b=c$
$a c=\frac{b}{2}$
Solution
Step 1: Differentiate the magnetic flux equation
The magnetic flux \(\phi \) is given by the relation \(\phi =at^{2}+bt+c\).
To find the time at which the flux is at its maximum value,
we need to find the point where its derivative with respect to time is equal to zero.
Taking the first derivative of the magnetic flux with respect to time,
\(t\):\(\frac{d\phi }{dt}=\frac{d}{dt}(at^{2}+bt+c)=2at+b\)
Step 2: Determine the relationship between the coefficients
The problem states that the magnetic flux reaches its maximum value at \(t=2\mathrm{~s}\).
At this point, the derivative of the flux with respect to time is zero. \(\frac{d\phi }{dt}=0\)Substitute \(t=2\) into the derivative equation:\(2a(2)+b=0\)\(4a+b=0\)
This equation relates the coefficients \(a\) and \(b\).
The coefficient \(c\) is a constant term and does not affect the time at which the flux is a maximum. Answer: The relation between \(a\), \(b\), and \(c\) is \(4a+b=0\).