UY1: Applications Of Ampere’s Law

Ampere’s Law can be used to simplify problems with a certain symmetry. Example: Magnetic Field Inside A Long Cylindrical Conductor A cylindrical conductor with radius R carries a current I. The current is uniformly distributed over the cross-sectional area of the conductor. Find the magnetic field as a function of the distance r from the conductor axis for points both …

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UY1: Ampere’s Law

Consider the magnetic field caused by a long, straight conductor carrying a current I: $$B = \frac{\mu_{0} I}{2 \pi r}$$ Let’s consider $\oint \vec{B}.d\vec{l}$ of the very same magnetic field. $$\begin{aligned} \oint \vec{B}.d\vec{l} &= \int\limits_{0}^{2 \pi} Br \, d \theta \\ &= Br \int\limits_{0}^{2 \pi} \, d\theta \\ &= B \times 2 \pi r \end{aligned}$$ Notice that we can substitute …

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UY1: Magnetic Field Of A Circular Current Loop

Consider a circular conductor with radius a that carries a current I. Find the magnetic field at point P on the axis of the loop, at a distance x from the center. From Biot-Savart Law: $$\vec{B} = \frac{\mu_{0}}{4\pi} \int \frac{I}{r^{2}} \, d\vec{l} \times \hat{r}$$ Due to the symmetry of the current loop, the $\vec{B_{y}}$ at point P due to the …

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UY1: Magnetic Field & Force Between Parallel Conductors

Magnetic Field Of Two Wires Figure shows an end view of two long, straight, parallel wires perpendicular to the xy-plane, each carrying a current I but in opposite directions. Find the magnitude and direction of $\vec{B}$ at point $P_{1}$. Find the magnitude and direction of $\vec{B}$ at any point on the x-axis to the right of wire 2 in terms …

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UY1: Magnetic Field Of A Straight Current Carrying Conductor

  Consider a straight conductor with length 2a carrying a current I. Find the magnetic field at point P which is located at a distance x from the conductor on its perpendicular bisector. From Biot-Savart Law, we have: $$\vec{B} = \frac{\mu_{0}}{4 \pi} \int \frac{I}{r^{2}} \, d\vec{l} \times \hat{r}$$ First, we will need to work out what is $d\vec{l} \times \hat{r}$. …

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UY1: Magnetic Field Of A Current Element

The magnetic field caused by a short segment $d\vec{l}$ of a current-carrying conductor can be obtained by a short derivation: We know that the magnetic field of a single point charge q moving with a constant velocity $\vec{v}$ is given by: $$B = \frac{\mu_{0}}{4 \pi} \frac{|q|v\sin{\phi}}{r^{2}}$$ Hence, $$dB = \frac{\mu_{0}}{4 \pi} \frac{|dQ|v_{d}\sin{\phi}}{r^{2}}$$ , where dQ is the element of charge in the …

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UY1: Magnetic Field Of A Moving Charge

The magnetic field of a single point charge q moving with a constant velocity $\vec{v}$ is given by: $$\vec{B} = \frac{\mu_{0}}{4 \pi} \frac{q}{r^{2}} \vec{v} \times \hat{r}$$ OR $$B = \frac{\mu_{0}}{4 \pi} \frac{|q|v\sin{\phi}}{r^{2}}$$ ,where $\mu_{0} = 4 \pi \times 10^{-7} \, \text{T m A}^{-1}$ The direction of $\vec{B}$ is perpendicular to the plane containing the line from source point to field point …

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