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Clarification about the problem "Danger? I laugh at danger!"

In this problem, when we consider rotation about the IAR, then it becomes as follows -

I1

I1

I2

I2

\[\begin{align} \mathrm{d}B &=\dfrac{\mu_o}{4\pi} \dfrac{\mathrm{d}q}{\mathrm{d}t} \dfrac{\vec{\mathrm{d}l} \times \vec{a}}{|\vec{a}|^3}\\ &=\dfrac{\mu_o}{4\pi}\mathrm{d}q \dfrac{2vr\sin(\theta/2) d}{(r^2+d^2)^{3/2}} \end{align}\]

Because \(\left|\vec{\left(\dfrac{\mathrm{d}l}{\mathrm{d}t}\right)}\right|=2v\sin (\theta/2)\) and \( \left|\vec{\left(\dfrac{\mathrm{d}l}{\mathrm{d}t}\right)} \times \vec{a}\right| = d\)

Now, \(\mathrm{d}Q=\dfrac{Q}{2\pi}\mathrm{d}\theta\).

So, \[\begin{align} B &=\dfrac{\mu_o}{4\pi^2} \dfrac{Qvd}{(r^2+d^2)^{3/2}} \int_{0}^{2\pi} \sin (\theta/2) \mathrm{d}\theta\\ &=\boxed{\dfrac{\mu_o Qvd}{\pi^2 (r^2+d^2)^{3/2}}} \end{align}\]

Note by Pratik Shastri
2 years, 2 months ago

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Hi Pratik,

Can you explain where the \(\sin\theta/2\) originates in your derivation? On the left hand side you seem to have a scalar \(B\) field (one component), while on the right you have a vector expression. When I calculate the components, I have some that cancel when integrated around the circle, and a \(z\)-component that survives. The \(z\)-component of the cross product gives \(\left(\vec{a}\times d\vec{l}\right)_z = r^2d\theta\). Josh Silverman Staff · 2 years, 2 months ago

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@Josh Silverman Yes, I found my mistake, I didn't check the \(x-y\) components.. but still why will do they cancel out? The top part of the wheel moves faster than the bottom part, right? Pratik Shastri · 2 years, 2 months ago

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@Pratik Shastri Are you thinking in the reference frame of the wheel or the frame of the ground? Josh Silverman Staff · 2 years, 2 months ago

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@Josh Silverman The ground. Pratik Shastri · 2 years, 2 months ago

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@Josh Silverman \(\dfrac{\mathrm{d}l}{\mathrm{d}t}\) is the velocity of the infinitesimally small element \(\mathrm{d}\theta\). The distance of this element from the instantaneous axis of rotation can be found using simple geometry. It comes out to be \(2r\sin (\theta/2)\). Also, \(\vec{\mathrm{d}l}\times \vec{a}\) is simply \(d\) as \(\vec{\mathrm{dl}}\) is always in the plane of the ring. Pratik Shastri · 2 years, 2 months ago

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