Thanks again! One last question though. Between which two points exactly is the potential difference ##-\frac{d\phi}{dt}## in cases such as these where the magnetic field varies with time. ?
Ahh thanks ! I knew something was wrong with my integral. But using ##x = \frac{L}{2} \sec\phi## and thus ##dx = \frac{L}{2}\sec \phi \tan \phi## would lead the integral being equal to ##-B\frac{L}{4} \sec \phi##. Putting in the limits I would get zero...
∫E.dl along the radii is zero. The area enclosed is ##\frac{L^2}{4}##. I am guessing this gives ∫E.dl along the groove, although I don't see how.
Anyway I too, noticed something. ##E \cos(\phi)##, where ##\phi## is the angle I marked in my Paint diagram/scribble but forgot to label. So there is...
So ##\pi r^2 B ## is the potential difference when ##E## is integrated along a circular path. Assuming magnitude of E to be constant along the circular path ##2\pi r## times ##E## is ##\pi r^2 B##. So ##E## is ##\frac{r}{2}B##. Since you mentioned that the induced field is non-conservative, I...
I don't really understand between which points exactly is the potential difference ##-\frac{d\phi}{dt}##. I can only think of it as the potential difference between the point and it's rotation by ##2\pi## radians, which seems nonsensical. But thinking about it this way I find that since the two...
Homework Statement
Uniform magnetic field ##= Bt## exist in cylindrical region of radius ##R## is pointing into the plane of figure (as shown in figure). A frictionless groove of length ##L## is fixed symmetrically from the centre O at a distance of ##\frac{L}{2}## .A charged particle of mass...
So, when ##cosx = -\frac{1}{2}## the equation is satisfied irrespective of the value of ##r## and so I get a line since r can vary for that particular value of x? Does this make sense? Thanks though, I just didn't know what to do with the equation I had.
Homework Statement
$$z^2 + z|z| + |z|^2=0$$
The locus of ##z## represents-
a) Circle
b) Ellipse
c) Pair of Straight Lines
d) None of these
Homework Equations
##z\bar{z} = |z|^2##
The Attempt at a Solution
Let ##z = r(cosx + isinx)##
Using this in the given equation
##r^2(cos2x + isin2x) +...
Homework Statement
Let ##g(x) = \int_0^xf(t) dt## where ##f## is such that ##\frac{1}{2} \leq f(t) \leq 1## for ##t \in [0,1]## and ##\frac{1}{2} \geq f(t) \geq 0## for ##t \in (1,2]##. Then ##g(2)## belongs to interval
A. ##[\frac{-3}{2}, \frac{1}{2}]##
B. ##[0, 2)##
C. ##(\frac{3}{2}...
Come to think of it, I don't really have a justification that the sphere will move tangentially to the arc at the point it leaves, save for an analogy with the case of uniform circular motion, which is certainly not happening here, but if not tangential, then which direction would the sphere...