Question

Two concentric co-planar circular Loops of radii \(\mathrm{r}_{1}\) and \(\mathrm{r}_{2}\) carry currents of respectively \(\mathrm{I}_{1}\) and \(\mathrm{I}_{2}\) in opposite directions. The magnetic induction at the centre of the Loops is half that due to \(\mathrm{I}_{1}\) alone at the centre. If \(\mathrm{r}_{2}=2 \mathrm{r}_{1}\) the value of \(\left(\mathrm{I}_{2} / \mathrm{I}_{1}\right)\) is (a) 2 (b) \(1 / 2\) (c) \(1 / 4\) (d) 1

Short answer

The value of \(\frac{I_2}{I_1}\) is 1, which corresponds to the answer (d) 1.

Step-by-step solution

Calculate the magnetic field at the center of loop 1

Since loop 1 carries \(I_1\) current and has a radius of \(r_1\), the magnetic field at the center of loop 1 can be calculated by: \[B_1 = \frac{\mu_0 I_1}{2r_1}\]

Calculate the magnetic field at the center of loop 2

Since loop 2 carries \(I_2\) current and has a radius of \(r_2\), the magnetic field at the center of loop 2 can be calculated by: \[B_2 = \frac{\mu_0 I_2}{2r_2}\] Given that the currents are in opposite directions, we need to subtract the magnetic fields.

Given condition

According to the given condition, the magnetic induction at the center of the loops(B) is half that due to \(I_1\) alone at the center. Therefore, \[B = B_1 - B_2 = \frac{1}{2} B_1\]

Substitute the values of B_1 and B_2

Now, replace the expressions of \(B_1\) and \(B_2\) that we found in Step 1 and Step 2 into the equation and solve for \(\frac{I_2}{I_1}\): \[\frac{\mu_0 I_1}{2r_1} - \frac{\mu_0 I_2}{4r_1} = \frac{1}{2} \frac{\mu_0 I_1}{2r_1}\]

Solve for I_2/I_1

Simplify the equation and solve for \(\frac{I_2}{I_1}\): \[\frac{I_1 - I_2 / 2}{2} = \frac{I_1}{4}\] \[I_1 - \frac{I_2}{2} = \frac{I_1}{2}\] \[I_1 - \frac{I_1}{2} = \frac{I_2}{2}\] \[\frac{I_2}{I_1} = 1\] The value of \(\frac{I_2}{I_1}\) is 1, which corresponds to the answer (d) 1.

Key concepts

Ampere's Law

Ampere's Law is a fundamental principle in electromagnetism that links the magnetic field in a loop to the electric current flowing through it. It is expressed mathematically as:\[ \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{enc} \]where:

• \(\mathbf{B}\) is the magnetic field,
• \(d\mathbf{l}\) is a differential element of the loop,
• \(\mu_0\) is the permeability of free space,
• \(I_{enc}\) is the current enclosed by the loop.

In simple terms, Ampere's Law allows us to calculate the magnetic field created by a given current distribution. When dealing with symmetrical arrangements, like concentric loops, it simplifies the computation of the magnetic field.
Here, Ampere's Law helps us understand how the current \(I_1\) and \(I_2\) affect the magnetic fields \(B_1\) and \(B_2\) at their center. It is crucial because it provides insights and initial checkpoints when analyzing the configuration's magnetic properties.

Biot-Savart Law

The Biot-Savart Law is another key concept in electromagnetism. It gives the magnetic field produced by a short segment of current-carrying wire. The principle is expressed as:\[ d\mathbf{B} = \frac{\mu_0}{4\pi} \frac{I \, d\mathbf{l} \times \mathbf{r}}{r^3} \]where:

• \(d\mathbf{B}\) is a small magnetic field contribution,
• \(I\) is the current,
• \(d\mathbf{l}\) is the length element of the wire,
• \(\mathbf{r}\) is the distance vector from the wire to the point of interest,
• \(\times\) denotes the cross product.

The Biot-Savart Law is particularly useful for finding the magnetic field of current distributions that are not highly symmetric. In this case, it can explain the contribution of each part of the loop to the overall magnetic field at the center.
In the exercise, it assists in understanding how each loop individually contributes to the magnetic field experienced at the center, using the given current magnitudes and loop radii.

Electromagnetism

Electromagnetism is a branch of physics focusing on the interaction between electric currents and magnetic fields. Central to this concept is the way currents influence magnetic fields and vice versa.

• Electric currents can create magnetic fields as shown by both Ampere's and Biot-Savart Laws.
• These laws help calculate magnetic fields in various geometries.
• Fields and currents interact in predictable ways as seen in configurations like concentric loops.

In practical terms, electromagnetism explains a wide range of phenomena from why magnets stick to metal objects to how electrical motors operate.
In this specific exercise, electromagnetism helps us understand the interaction of magnetic fields from each loop. By analyzing how these fields combine or cancel out, we can determine the resultant magnetic field at the center, applying the relationship between magnetic fields, currents, and geometries.