A thin circular coil of radius r=15 cm contains N=3 turns of Nichrome wire. A small compass is p

step 1 Hi, the given problem there are two concentric coplanar horizontal circular coils like this. The

A thin circular coil of radius r=15 cm contains N=3 turns of...

A thin circular coil of radius $r=15 \mathrm{cm}$ contains $N=3$ turns of Nichrome wire. A small compass is placed at the center of the coil, as shown in Figure $17.92 .$ With the battery disconnected, the compass needle points to the right, in the plane of the coil. Assume that the horizontal component of the Earth's magnetic field is about $B_{\text {Earth }} \approx 2 \times 10^{-5} \mathrm{T}$. When the battery is connected, a current of 0.25 A runs through the coil. Predict the deflection of the compass needle. If you have to make any approximations, state what they are. Is the deflection outward or inward as seen from above? What is the magnitude of the deflection?

A thin circular coil of radius $r=15 \mathrm{cm}$ contains $N=3$ turns of Nichrome wire. A small compass is placed at the center of the coil, as shown in Figure $17.92 .$ With the battery disconnected, the compass needle points to the right, in the plane of the coil. Assume that the horizontal component of the Earth's magnetic field is about $B_{\text {Earth }} \approx 2 \times 10^{-5} \mathrm{T}$.
When the battery is connected, a current of 0.25 A runs through the coil. Predict the deflection of the compass needle. If you have to make any approximations, state what they are. Is the deflection outward or inward as seen from above? What is the magnitude of the deflection?

Key Concepts

Right-Hand Rule

The right-hand rule is a mnemonic used to determine the direction of the magnetic field produced by a current. By orienting the thumb in the direction of the conventional current, the curl of the fingers shows the direction of the magnetic field lines encircling the wire or loop. This rule helps determine whether the field from the coil will add to or subtract from the horizontal component of the Earth's magnetic field at the compass's location.

Compass Needle Deflection

A compass needle aligns along the direction of the magnetic field it experiences. When an additional magnetic field, such as that from a current-carrying coil, is applied, the needle deflects from its original orientation, which is set by the Earth's field. The deflection angle can be calculated using trigonometric relationships based on the ratio of the applied magnetic field to the Earth's magnetic field.

Superposition of Magnetic Fields

When multiple magnetic fields are present, their net effect is the vector sum of the individual fields. In this context, the field produced by the coil combines with the Earth's ambient magnetic field. The compass needle, which aligns with the total magnetic field, deflects by an angle determined by the relative magnitudes and directions of these individual fields.

Magnetic Field of a Current Loop

A current loop generates a magnetic field that is strongest at the center of the loop. The magnitude of this field, derived from the Biot–Savart law, is given by B = (??NI) / (2r) for a thin circular coil with N turns carrying a current I, where r is the radius. This concept underpins the calculation of the magnetic field produced by the coil in many electromagnetic applications.

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