Answer
Verified
377.1k+ views
Hint: Considering a small element at a distance xx of small width dx.dx.
Magnetic moment of the small element is- \[dm = 2\pi (lqdx)\omega \pi r^2\].The magnetic moment of the whole rod can be found by integrating that value over the whole length of the rod.
Complete answer:
Let a rod be given and is fixed at its mid-point. So here, we have to find the magnetic moment of the rod system.
Let's go to ‘x’ distance and take an element named ‘dx’.
Then dm \[ = \] idA
In l length charge is q; I length charge is
\[
\\
\dfrac{{q\pi f{l^2}}}{{12}} \\
\]; dx length charge is \[\dfrac{q}{l} \times dx\]. \[\int { = qf} \]
\[
\therefore dm = idA \\
= \dfrac{q}{l}dx \\
\]
Since, \[i = \dfrac{q}{T}\] we can write this in frequency terms as \[l = qf\]
Then, if the area is \[\pi {r^2}\] and is rotating along with the ‘x’ radius hence the equation will be \[dm = \dfrac{q}{l}dxf{\pi ^2}\]. Now here we got the magnetic moment due to the small element. If we have to find the entire magnetic moment then we will integrated the dm function and the equation will now become as :
\[{\int {dm = \int {\dfrac{q}{{{l_{_{\dfrac{{ - l}}{2}}}}}}} } ^{\dfrac{l}{2}}} \times f\pi {x^2}dx\]
\[m = \dfrac{q}{l} \times \pi f{[\dfrac{{{x^3}}}{3}]^{\dfrac{l}{2}}}_{\dfrac{{ - l}}{2}}\]
After this we get,
\[
= \dfrac{{q\pi f}}{l} \times [\dfrac{{{l^3}}}{{24}} - (\dfrac{{ - {l^{}}}}{{24}})] \\
= \dfrac{{q\pi f}}{l}(\dfrac{{2{l^3}}}{{24}}) \\
= \dfrac{{q\pi f{l^2}}}{{12}} \\
\]
Hence, the correct answer is \[
\\
\dfrac{{q\pi f{l^2}}}{{12}} \\
\].
Note: The magnetic moment is the strength and orientation of a magnet or any other object that produces a magnetic field. The rod sweeps a circle of radius \[\dfrac{1}{2}\].
Magnetic moment of the small element is- \[dm = 2\pi (lqdx)\omega \pi r^2\].The magnetic moment of the whole rod can be found by integrating that value over the whole length of the rod.
Complete answer:
Let a rod be given and is fixed at its mid-point. So here, we have to find the magnetic moment of the rod system.
Let's go to ‘x’ distance and take an element named ‘dx’.
Then dm \[ = \] idA
In l length charge is q; I length charge is
\[
\\
\dfrac{{q\pi f{l^2}}}{{12}} \\
\]; dx length charge is \[\dfrac{q}{l} \times dx\]. \[\int { = qf} \]
\[
\therefore dm = idA \\
= \dfrac{q}{l}dx \\
\]
Since, \[i = \dfrac{q}{T}\] we can write this in frequency terms as \[l = qf\]
Then, if the area is \[\pi {r^2}\] and is rotating along with the ‘x’ radius hence the equation will be \[dm = \dfrac{q}{l}dxf{\pi ^2}\]. Now here we got the magnetic moment due to the small element. If we have to find the entire magnetic moment then we will integrated the dm function and the equation will now become as :
\[{\int {dm = \int {\dfrac{q}{{{l_{_{\dfrac{{ - l}}{2}}}}}}} } ^{\dfrac{l}{2}}} \times f\pi {x^2}dx\]
\[m = \dfrac{q}{l} \times \pi f{[\dfrac{{{x^3}}}{3}]^{\dfrac{l}{2}}}_{\dfrac{{ - l}}{2}}\]
After this we get,
\[
= \dfrac{{q\pi f}}{l} \times [\dfrac{{{l^3}}}{{24}} - (\dfrac{{ - {l^{}}}}{{24}})] \\
= \dfrac{{q\pi f}}{l}(\dfrac{{2{l^3}}}{{24}}) \\
= \dfrac{{q\pi f{l^2}}}{{12}} \\
\]
Hence, the correct answer is \[
\\
\dfrac{{q\pi f{l^2}}}{{12}} \\
\].
Note: The magnetic moment is the strength and orientation of a magnet or any other object that produces a magnetic field. The rod sweeps a circle of radius \[\dfrac{1}{2}\].
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Which are the Top 10 Largest Countries of the World?
Give 10 examples for herbs , shrubs , climbers , creepers
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Change the following sentences into negative and interrogative class 10 english CBSE
How do you graph the function fx 4x class 9 maths CBSE
Write a letter to the principal requesting him to grant class 10 english CBSE