Question

Consider 3 identical conducting loops kept parallel to each other such that their axes coincide. Forloop (1), loop (2) and loop (3) self inductance of each loop is L. Current in loop (1) is I₁, in loop (2) I₂ and in loop (3)I₃, rate of change in current in them at certain instant is $\frac{dI_1}{dt}$, $\frac{dI_2}{dt}$, $\frac{dI_3}{dt}$ respectively. Their mutual inductance is Mᵢⱼ(i.e. for 1 & 2 its M- 12) current in loop (1) and (3) is in same sense and in (2) its in opposite sense:

Answer 1

Missing Question

Answer 2

The question provided is incomplete. It describes a system of three identical conducting loops and provides information about their self-inductance, mutual inductance, currents, and rates of change of currents, as well as the relative sense of the currents. However, it does not state what needs to be calculated or determined.

Assuming the question is intended to ask about the induced electromotive force (EMF) in one or more of the loops, or perhaps the total magnetic energy stored in the system, we can set up the relevant expressions.

Let the self-inductance of each loop be $L$. Let the mutual inductance between adjacent loops (1 and 2, or 2 and 3) be $M$. Since the loops are identical and presumably equally spaced, $M_{12} = M_{23} = M$. Let the mutual inductance between loops 1 and 3 be $M'$. Since loop 3 is further from loop 1 than loop 2 is, we expect $M' < M$. We assume $L, M, M'$ are positive constants.

Let $I_1, I_2, I_3$ be the currents in loops 1, 2, and 3 respectively. We define a common positive direction for current in all loops (e.g., clockwise when viewed from a certain side). With this convention, the mutual inductances $M$ and $M'$ are positive. The problem states that the current in loop (1) and (3) is in the same sense, and in (2) it is in the opposite sense. This means that at the instant considered, the sign of $I_1$ is the same as the sign of $I_3$, and the sign of $I_2$ is opposite to the sign of $I_1$ and $I_3$. For example, if $I_1 > 0$ and $I_3 > 0$, then $I_2 < 0$. This information about the instantaneous values of the currents is relevant for calculating quantities like magnetic field or force, but not directly for the induced EMF which depends on the rates of change of currents.

The magnetic flux linkage $\Phi_i$ in loop $i$ is given by the sum of the flux due to its own current and the fluxes due to the currents in the other loops: $\Phi_1 = L I_1 + M_{12} I_2 + M_{13} I_3 = L I_1 + M I_2 + M' I_3$ $\Phi_2 = M_{21} I_1 + L I_2 + M_{23} I_3 = M I_1 + L I_2 + M I_3$ $\Phi_3 = M_{31} I_1 + M_{32} I_2 + L I_3 = M' I_1 + M I_2 + L I_3$

The induced EMF $E_i$ in loop $i$ is given by Faraday's law of induction, $E_i = - \frac{d\Phi_i}{dt}$: $E_1 = - \frac{d}{dt}(L I_1 + M I_2 + M' I_3) = - (L \frac{dI_1}{dt} + M \frac{dI_2}{dt} + M' \frac{dI_3}{dt})$ $E_2 = - \frac{d}{dt}(M I_1 + L I_2 + M I_3) = - (M \frac{dI_1}{dt} + L \frac{dI_2}{dt} + M \frac{dI_3}{dt})$ $E_3 = - \frac{d}{dt}(M' I_1 + M I_2 + L I_3) = - (M' \frac{dI_1}{dt} + M \frac{dI_2}{dt} + L \frac{dI_3}{dt})$

These equations describe the induced EMF in each loop at the given instant, based on the self and mutual inductances and the rates of change of currents.

The total magnetic energy $U$ stored in the system of three coupled inductors is given by: $U = \frac{1}{2} L_{11} I_1^2 + \frac{1}{2} L_{22} I_2^2 + \frac{1}{2} L_{33} I_3^2 + M_{12} I_1 I_2 + M_{13} I_1 I_3 + M_{23} I_2 I_3$ Substituting the given values $L_{11} = L_{22} = L_{33} = L$, $M_{12} = M_{23} = M$, and $M_{13} = M'$: $U = \frac{1}{2} L I_1^2 + \frac{1}{2} L I_2^2 + \frac{1}{2} L I_3^2 + M I_1 I_2 + M' I_1 I_3 + M I_2 I_3$ The condition on the sense of the currents means that at the given instant, $I_1$ and $I_3$ have the same sign, and $I_2$ has the opposite sign. This implies that the products $I_1 I_2$ and $I_2 I_3$ are negative, while $I_1 I_3$ is positive. Incorporating this into the energy formula, if $I_1, I_2, I_3$ represent the signed values: $U = \frac{1}{2} L (I_1^2 + I_2^2 + I_3^2) + M I_1 I_2 + M' I_1 I_3 + M I_2 I_3$

If the question intended to ask for one of these quantities, the expressions above provide the solution in terms of the given parameters. Without a specific question, a definitive numerical or symbolic answer cannot be provided.

Assuming this is a multiple choice question and the options are missing, the likely intended question might be related to one of the induced EMFs or the total energy. The expressions derived above are the standard formulas for such a system.

Let's assume the question asks for the induced EMF in loop 2, as it is centrally placed and influenced by both other loops. The induced EMF in loop 2 is $E_2 = - (M \frac{dI_1}{dt} + L \frac{dI_2}{dt} + M \frac{dI_3}{dt})$.

If the question was, for example, "Write the expression for the induced EMF in loop 2", then the answer would be $E_2 = - (M \frac{dI_1}{dt} + L \frac{dI_2}{dt} + M \frac{dI_3}{dt})$.

If the question was, for example, "Write the expression for the total magnetic energy stored in the system at the given instant", then the answer would be $U = \frac{1}{2} L (I_1^2 + I_2^2 + I_3^2) + M I_1 I_2 + M' I_1 I_3 + M I_2 I_3$.

Since the problem only provides the setup and parameters without asking a specific question, the solution involves listing the relevant physics principles and equations for such a system. The most relevant quantities when dealing with changing currents and inductances are magnetic flux linkage, induced EMF, and stored magnetic energy.

Given the context of JEE/NEET, questions typically ask for a specific value or expression. Without the question itself, we can only provide the fundamental equations governing the system. The information about the sense of currents is likely important for a specific calculation (like energy or force) at that instant, but the EMF equations derived are generally valid for any instantaneous values of currents and their rates of change, provided $I_1, I_2, I_3$ are consistently signed variables.

Assuming the most probable type of question given the provided data is about induced EMF, we present the expressions for induced EMF in each loop.

The final answer is the set of equations for the induced EMF in each loop or the total magnetic energy, depending on what the incomplete question was asking for. Without the actual question, we cannot select a specific final answer. We provide the expressions for induced EMF as they directly use the rate of change of currents mentioned in the problem.

Induced EMF in loop 1: $E_1 = - (L \frac{dI_1}{dt} + M \frac{dI_2}{dt} + M' \frac{dI_3}{dt})$ Induced EMF in loop 2: $E_2 = - (M \frac{dI_1}{dt} + L \frac{dI_2}{dt} + M \frac{dI_3}{dt})$ Induced EMF in loop 3: $E_3 = - (M' \frac{dI_1}{dt} + M \frac{dI_2}{dt} + L \frac{dI_3}{dt})$ where $M$ is the mutual inductance between adjacent loops (1-2 and 2-3) and $M'$ is the mutual inductance between loops 1 and 3.

The final answer is $\boxed{Missing Question}$.