SATHEE: Magnetics 3 Question 12

Magnetics 3 Question 12

12. A straight segment $O C$ (of length $L$ ) of a circuit carrying a current $I$ is placed along the $x$ -axis . Two infinitely long straight wires $A$ and $B$ , each extending from $z = - \infty$ to $+ \infty$ , are fixed at $y = - a$ and $y = + a$ respectively, as shown in the figure. If the wires $A$ and $B$ each carry a current $I$ into the plane of the paper, obtain the expression for the force acting on the segment $O C$ . What will be the force on $O C$ if the current in the wire $B$ is reversed?

$(1992, 10 M)$

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Answer:

Correct Answer: 12. (a) $3 A$ , perpendicular to paper outwards

(b) $13 \times 10^{- 7} T$

Solution:

(a) Let us assume a segment of wire $O C$ at a point $P$ , a distance $x$ from the centre of length $d x$ as shown in figure.

Magnetic field at $P$ due to current in wires $A$ and $B$ will be in the directions perpendicular to $A P$ and $B P$ respectively as shown.

$| B | = \frac{μ_{0}}{2 π} \frac{I}{A P}$

Therefore, net magnetic force at $P$ will be along negative $y$ -axis as shown

$B_{net} = 2 | B | \cos θ = 2 \frac{μ_{0}}{2 π} \frac{I}{A P} \frac{x}{A P}$

$\begin{aligned} B_{net} = \frac{μ_{0}}{π} \frac{I \cdot x}{(A P)^{2}} \\ B_{net} = \frac{μ_{0}}{π} \cdot \frac{I x}{(a^{2} + x^{2})} \end{aligned}$

Therefore, force on this element will be

$d F = I \frac{μ_{0}}{π} \frac{I x}{a^{2} + x^{2}} d x (in negative z -direction)$

$∴$ Total force on the wire will be

$\begin{aligned} F & = \int_{x = 0}^{x = L} d F = \frac{μ_{0} I^{2}}{π} \int_{0}^{L} \frac{x d x}{x^{2} + a^{2}} \\ = \frac{μ_{0} I^{2}}{2 π} \ln \frac{L^{2} + a^{2}}{a^{2}} (in negative z -axis) \end{aligned}$

Hence, $F = - \frac{μ_{0} I^{2}}{2 π} \ln \frac{L^{2} + a^{2}}{a^{2}} \hat{k}$

(b) When direction of current in $B$ is reversed, net magnetic field is along the current. Hence, force is zero.

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Magnetics 3 Question 12

Answer:

इंजीनियरिंग की तैयारी

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एनसीईआरटी व्यायाम वीडियो समाधान

Magnetics 3 Question 12

Answer:

इंजीनियरिंग की तैयारी

चिकित्सा तैयारी

एनसीईआरटी पुस्तकें और समाधान

महत्वपूर्ण संसाधन

अन्य संसाधन

पाठ्यक्रम

परीक्षा मंच

नमूना मॉक टेस्ट

सदस्यता

एनसीईआरटी व्यायाम वीडियो समाधान

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